NCERT Exemplar for Class 11 Maths Chapter 14 - Mathematical Reasoning (Book Solutions)

Examples:

1. Which of the following sentences is a statement.

(i) New Delhi is in India.

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

The sentence ‘New Delhi is in India’ is true. Therefore, it is a statement.

(ii) Every rectangle is a square.

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

The sentence ‘Every rectangle is a square’ is true. Therefore, it is a statement.

(iii) Close the door.

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

Since, the sentence ‘Close the door’ is an order. Therefore, it is not a statement.

(iv) How old are you?

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false. Since, the sentence ‘How old are you?’ is a question. Therefore, it is not a statement.

(v) x is a natural number.

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false. Since, the truth or falsity of sentence ‘x is a natural number’ depends on x. Therefore, it is not a statement.

2. The statements:

Ans: “$2$ is an even number”

“A square has all its sides equal” and,

“Chandigarh is the capital of Haryana” are all simple statements.

3. The statement “$11$is both an odd and prime number”.

Ans: Since, the statement is broken into two statements,

“ $11$ is both an odd” and $11$ is a prime number”.

Therefore, it is a compound statement.

4. Form the conjunction of the following simple statements

p: Dinesh is a boy.

q: Nagma is a girl.

Ans: Given: Simple statements,

p: Dinesh is a boy.

q: Nagma is a girl.

We know that, Conjunction of simple statements is obtained by using connector ‘AND’ between the statements.

The conjunction of two simple statements is represented by ${\text{p }} \wedge {{ q}}{\text{.}}$

Therefore, the conjunction of the given statement is given by,

${{p }} \wedge {{ q :}}$ Dinesh is a boy and Nagma is a girl.

5. Translate the following statement into symbolic form

“Jack and Jill went up the hill”.

Ans: Given: The statement “Jack and Jill went up the hill”.

We know that Symbolic forms use logical connectors to represent the statement logically. First, rewrite the statement. Then use symbolic form between the statements.

First, break the statement in ${\text{p and q}}$as,

${{p :}}$ Jack went up the hill.

${{q :}}$ Jill went up the hill.

Now, rewrite the given statement such that the logical connector is used in the statement.

The statement can be written as, “Jack went up the hill and Jill went up the hill”.

Therefore, the statement in symbolic form is given by ${{p }} \wedge {{ q}}{\text{.}}$

6. Write the truth value of each of the following four statements:

(i) Delhi is in India and $2  +  3  =  6.$

Ans: We know that,

Use the truth value of conjunction ${\text{p }} \wedge {\text{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$

According to the truth value of conjunction ${{p }} \wedge {{ q,}}$ the statement possess truth value ${{F}}$ as the truth value of the statement is ${{F}}{\text{.}}$

(ii) Delhi is in India and $2  +  3  =  5.$

Ans: We know that,

Use the truth value of conjunction ${{p }} \wedge {{ q}}$of two simple statements ${\text{p and q}}{\text{.}}$

According to the truth value of conjunction ${{p }} \wedge {{ q,}}$ the statement possess truth value ${{T}}$ as both the statement “Delhi is in India” and “${{2  +  3  =  5}}{\text{.}}$” possess truth value ${{T}}{{.}}$

(iii) Delhi is in Nepal and $2  +  3  =  5.$

Ans: We know that,

Use the truth value of conjunction ${{p }} \wedge {{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$ The truth value of the statements is ${{F}}{\text{.}}$ As truth value of both statements is ${{F}}{\text{.}}$

(iv) Delhi is in Nepal and $2  +  3  =  6.$

Ans: We know that,

Use the truth value of conjunction ${{p }} \wedge {{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$

The truth value of the statements is  ${\text{F}}{\text{.}}$ As truth value of both statements is ${{F}}{\text{.}}$

7. Form the disjunction of the following simple statements:

p: The sun shines.

q: It rains.

Ans: Given: Simple statements.

p: The sun shines.

q: It rains.

We know that, Disjunction of simple statements is obtained by using connector ‘OR’ between the statements.

The disjunction of two simple statements is represented by ${{p }} \vee {{ q}}{\text{.}}$

Therefore, the conjunction of the given statement is given by,

${{p }} \vee {{ q :}}$ The sun shines or it rains.

8. Write the truth value of each of the following statements:

(i) India is in Asia or 2 + 2 = 4

Ans: We know that,

Use the truth value of disjunction ${{p }} \vee {{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$ If one of the sub-statement has truth value ${{T}}{\text{.}}$ Then, truth value of statement is ${{T}}{\text{.}}$

According to the truth value of disjunction ${{p }} \vee {{ q,}}$ the statement possess truth value ${{T}}{\text{.}}$ As both the sub-statement has truth value ${{T}}{\text{.}}$

(ii) India is in Asia or 2 + 2 = 5

Ans: We know that,

Use the truth value of disjunction ${{p }} \vee {{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$ If one of the sub-statement has truth value ${{T}}{\text{.}}$ Then, truth value of statement is ${{T}}{\text{.}}$

According to the truth value of disjunction ${{p }} \vee {{ q,}}$ the statement possess truth value ${{T}}{{.}}$ As at least one of the sub-statement has truth value ${{T}}{\text{.}}$

(iii) India is in Europe or 2 + 2 = 4

Ans: We know that,

Use the truth value of disjunction ${{p }} \vee {{ q}}$ of two simple statements ${\text{p and q}}{{.}}$ If one of the sub-statement has truth value ${{T}}{\text{.}}$ Then, truth value of statement is ${{T}}{\text{.}}$

According to the truth value of disjunction ${{p }} \vee {{ q,}}$ the statement possess truth value ${{T}}{\text{.}}$ As at least one of the sub-statement has truth value ${{T}}{\text{.}}$

(iv) India is in Europe or $2  +  2  =  5.$

Ans: We know that,

Use the truth value of disjunction ${{p }} \vee {{ q}}$ of two simple statements ${\text{p and q}}{{.}}$ If one of the sub-statement has truth value ${{T}}{\text{.}}$ Then, truth value of statement is ${{T}}{\text{.}}$

According to the truth value of disjunction ${{p }} \vee {{ q,}}$ the statement possess truth value ${{F}}{\text{.}}$ As both the sub-statement has truth value ${{F}}{\text{.}}$

9. Write the negation of the statement,

p: New Delhi is a city.

Ans: Given: Statement,

${{p :}}$ New Delhi is a city.

We know that,Negation of the statement changes the truth value ${{T \to F}}$ and ${{F \to T}}{\text{.}}$

The negation of the statement ${{p}}$ is represented by $~p$

Therefore, the negation of the given statement ${{p}}$ is given by,

$~{{ p :}}$ New Delhi is not a city.

10. Write the truth value of the negation of each of the following statements:

(i) p: Every square is a rectangle.

Ans: We know that, first, find the truth value of the given statements. Then, find the truth value of their negation.

The truth value of the statement p is T. Therefore, the truth value of the statement ${{~ p}}$ is ${{F}}{\text{.}}$

(ii) q: The earth is a star.

Ans: We know that, first, find the truth value of the given statements. Then, find the truth value of their negation.

The truth value of the statement q is F. Therefore, the truth value of the statement ${{~ q}}$ is ${{T}}{\text{.}}$

(iii) $r : 2  +  3  <  4$

Ans: We know that, first, find the truth value of the given statements. Then, find the truth value of their negation. The truth value of the statement r is F. Therefore, the truth value of the statement $\sim {{r}}$ is T.

11. Write the negation of each of the following conjunctions:

(a). Paris is in France and London is in England.

Ans: We know that, the conjunction of the statement ${{p }} \wedge {{ q}}$ is given by ${{~ (p }} \wedge {{ q)}}{\text{.}}$ First, find the sub-statement of the conjunction.

Let us consider the sub-statements of conjunction as,

${{p :}}$ Paris is in France, and

${{q :}}$ London is in England.

Now, taking negation of the sub-statements is given by,

${{~ p :}}$ Paris is not in France, and

${{~ q :}}$ London is not in England.

Therefore, negation of the given conjunction is given by,

${{~ (p }} \wedge {{ q) :}}$ Paris is not in France or London is not in England.

(b) $2  +  3  =  5 \text{ and }8  <  10.$

Ans: We know that, the conjunction of the statement ${{p }} \wedge {{ q}}$ is given by ${{~ (p }} \wedge {{ q)}}{\text{.}}$ First, find the sub-statement of the conjunction.

Let us consider the sub-statements of conjunction as,

${{p : 2  +  3  =  5}}$ and,

${{q : 8  <  10}}{{.}}$

Now, taking negation of the sub-statements,

${{~p : 2  +  3 }} \ne {{ 5}}$ and

${{~q : 8 \text{ not }  <  10}}{\text{.}}$

Therefore, negation of conjunction is given by,

$~(p \wedge q) : 2+3 \ne 5$ or 8 not  <  10

12. Write the negation of each of the following disjunction:

(a) Ram is in Class X or Rahim is in Class XII.

Ans: We know that, the negation of disjunction ${{~ (p }} \vee {{ q)}}$ can be obtained by conjunction of negation of ${{~ p and ~ q}}{\text{.}}$

Let us consider the sub-statements of disjunction as,

${{p :}}$ Ram is in class ${{X}}{\text{.}}$

${{q :}}$ Rahim is in class ${{XII}}{\text{.}}$

Now, taking negation of the sub-statements is given by,

${{~ p :}}$ Ram is not in class ${{X}}{\text{.}}$

${{~ q :}}$ Rahim is not in class ${{XII}}{\text{.}}$

Therefore, negation of the given disjunction is given by,

${{~ (p }} \vee {{ q) :}}$ Ram is not in class ${{X}}$ and Rahim is not in class ${{XII}}{\text{.}}$

(b) 7 is greater than 4 or 6 is less than 7.

Ans: We know that, the negation of disjunction ${{~ (p }} \vee {{ q)}}$ can be obtained by conjunction of negation of ${{~ p \text{ and }~ q}}{\text{.}}$

Let us consider the sub-statements of disjunction as,

${{p :}}$${{7}}$ is greater than ${{4}}{\text{.}}$

${\text{q :}}$${\text{6}}$ is less than ${{7}}$.

Now, taking negation of the sub-statements is given by,

${{~ p :}}$${{7}}$ is not greater than ${{4}}{\text{.}}$

${{~ q :}}$${{6}}$ is not less than ${{7}}$.

Therefore, negation of the given disjunction is given by,

${{~ (p }} \vee {{ q) :}}$${{7}}$ is not greater than ${{4 \text{and} 6}}$ is not less than ${{7}}$.

13 Each of the following statements is also a conditional statement.

(i) If $2+2=5$, then Rekha will get an ice-cream.

(ii) If you eat your dinner, then you will get dessert.

(iii) If John works hard, then it will rain today.

(iv) If $\mathrm{ABC}$ is a triangle, then $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=180^{\circ}$.

14. Express in English, the statement $p  \to  q,$ where

$p:$ it is raining today

$q : 2  +  3  >  4$

Ans: Given: The statement,

${{p :}}$ it is raining today

${{q : 2  +  3  >  4}}$

We know that, the conditional statement ${{p }} \to {{ q}}$ is expressed as if ${\text{p, then q}}{\text{.}}$

The required conditional statement ${{p }} \to {{ q}}$ can be expressed as,

${{p }} \to {{ q :}}$ “If it is raining today, then ${{2  +  3  >  4}}$”.

15. Write each of the following statements in its equivalent contrapositive form:

(i) If my car is in the repair shop, then I cannot go to the market.

Ans: We know that, the equivalent contrapositive of the conditional statement ${{p }} \to {{ q}}$ is given by ${{~ q }} \to {{ ~ p}}{\text{.}}$

For the given statements.

The equivalent contrapositive form of the statement can be given as,

${{~ q }} \to {{ ~ p :}}$ “If I can go to the market, then my car is not in the repair shop”.

(ii) If Karim cannot swim to the fort, then he cannot swim across the river.

Ans: We know that, the equivalent contrapositive of the conditional statement ${{p }} \to {{ q}}$ is given by ${{~ q }} \to {{ ~ p}}{\text{.}}$

For the given statements.

The equivalent contrapositive form of the statement can be given as,

${{~ q }} \to {{ ~ p :}}$ “If Karim can swim across the river, then he can swim to the fort”.

16. Write the converse of the following statements:

(i) If $x  <  y, \text{ then }x  +  5  <  y  +  5.$

Ans: We know that Converse is logically obtained by reverse of the sub-statements.

Here,

${{p : x  <  y}}$

${{q : x  +  5  <  y  +  5}}$

Converse of the statement is given by,

(ii) If $ABC$ is an equilateral triangle, then $ABC$ is an isosceles triangle.

Ans: We know that Converse is logically obtained by reverse of the sub-statements.

Here,

The converse of the statement is given by,

“If ${\text{ABC}}$ is an isosceles triangle, then ${\text{ABC}}$ is an equilateral triangle.”

17. Form the biconditional of the following statements:

$p :$ One is less than seven.

$q :$ Two is less than eight.

Ans: Given: Statement

${{p :}}$ One is less than seven.

${{q :}}$ Two is less than eight.

We know that, Bi-conditional statement is obtained by connecting conditional statements using if and only if.

The bi-conditional statement of the conditional statement is given by “One is less than seven, if and only if two is less than eight”.

18. Translate the following bi-conditional into symbolic form:

“$ABC$ is an equilateral triangle if and only if it is equiangular”.

Ans: Given: Bi-conditional statement.

We know that symbolic forms use logical connectors to represent the statement logically. First, rewrite the statement. Then use symbolic form between the statements.

Let the sub-statement as,

${{p : ABC}}$ is an equilateral triangle.

${{q : ABC}}$ is an equiangular triangle.

The statement in symbolic form is given by ${{p }} \leftrightarrow {{ q}}{\text{.}}$

Short Answer Type:

19. Which of the following statements are compound statements:

(i). “2 is both an even number and a prime number”

Ans: We know that, the statements which can be divided into two simple statements are called compound statements. The given statement can be broken into two statements. Therefore, it is a compound statement.

(ii).“$9$ is neither an even number nor a prime number”

Ans: We know that, the statements which can be divided into two simple statements are called compound statements. The given statement can be broken into two statements. Therefore, it is a compound statement.

(iii).“Ram and Rahim are friends”

Ans: We know that, the statements which can be divided into two simple statements are called compound statements. The given statement cannot be broken into two statements. Therefore, it is not a compound statement.

20. Identify the component statements and the connective in the following compound statements.

(a) It is raining or the sun is shining.

Ans: We know that, Component statements are the sub-statements in the compound statement. Connectives connect the two simple statements.

The component statements are given by,

${{p :}}$ It is raining.

${{q :}}$ The sun is shining.

The connective is “or”.

(b) 2 is a positive number or a negative number.

Ans: We know that, Component statements are the sub-statements in the compound statement. Connectives connect the two simple statements.

The component statements are given by,

${{p : 2}}$ is a positive number.

${{q : 2}}$ is a negative number.

The connective is ‘or’.

21. Translate the following statements in symbolic form:

(i) 2 and 3 are prime numbers.

Ans: We know that Symbolic forms use logical connectors to represent the statement logically. First, rewrite the statement. Then use symbolic form between the statements.

The statement can be written as,

“${{2}}$ is a prime number and ${{3}}$ is a prime number”.

${{p : 2}}$ is a prime number.

${{q : 3}}$ is a prime number.

The symbolic form of the statement is given by ${{p }} \wedge {{ q}}{\text{.}}$

(ii) Tigers are found in Gir forest or Rajaji national park.

Ans: We know that Symbolic forms use logical connectors to represent the statement logically. First, rewrite the statement. Then use symbolic form between the statements.

The statement can be written as,

“Tigers are found in Gir forest or Tigers are found in Rajaji national park”

${{p :}}$ Tigers are found in Gir forest.

${{q :}}$ Tigers are found in Rajaji national park.

The symbolic form of the statement is given by ${{p }} \vee {{ q}}{\text{.}}$

22. Write the truth value of each of the following statements.

(i) 9 is an even integer or $9  +  1$ is even.

Ans: We know that, use the truth value of disjunction ${{p }} \vee {{ q}}$ and conjunction ${{p }} \wedge {{ q}}$ of two simple statements ${{p \text{ and } q}}{{.}}$ Here, at least one component statement has truth value ${{T}}{\text{.}}$ Therefore, the truth value of statement is ${{T}}{\text{.}}$

(ii) $2  +  4  =  6 \text{ or } 2  +  4  =  7.$

Ans: We know that, use the truth value of disjunction ${{p }} \vee {{ q}}$ and conjunction ${{p }} \wedge {{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$ Here, at least one component statement has truth value ${{T}}{{.}}$ Therefore, the truth value of statement is ${{T}}{\text{.}}$

(iii) Delhi is the capital of India and Islamabad is the capital of Pakistan.

Ans: We know that, use the truth value of disjunction ${{p }} \vee {{ q}}$ and conjunction ${{p }} \wedge {{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$ Here, both the component statement has truth value ${{T}}{\text{.}}$ Therefore, the truth value of statement is ${{T}}{\text{.}}$

(iv) Every rectangle is a square and every square is a rectangle.

Ans: We know that, use the truth value of disjunction ${{p }} \vee {{ q}}$ and conjunction ${{p }} \wedge {{ q}}$ of two simple statements ${\text{p and q}}{\text{.}}$ Here, one component statement has truth value ${{F}}{\text{.}}$ Therefore, the truth value of statement is ${{F}}{\text{.}}$

(v) The sun is a star or the sun is a planet.

Ans: We know that, use the truth value of disjunction ${{p }} \vee {{ q}}$ and conjunction ${{p }} \wedge {{ q}}$ of two simple statements ${\text{p and q}}{{.}}$ Here, at least one component statement has truth value ${{T}}{\text{.}}$ Therefore, the truth value of statement is ${{T}}{\text{.}}$

23. Write negation of the statement:

“Everyone who lives in India is an Indian”.

Ans: Negation of the statement changes the truth value ${{T \to F}}$ and ${{F \to T}}{\text{.}}$

Let ${{p :}}$ Everyone who lives in India is an Indian.

The negation of the statement ${{p}}$ is given by,

${{~ p :}}$ It is false that everyone who lives in India is an Indian.

24. Write the negation of the following statements:

(a) p: All triangles are equilateral triangles. 

Ans: We know that, Negation of the statement changes the truth value ${{T \to F}}$ and ${{F \to T}}{\text{.}}$ The negation of the statement is given by,

${\text{~ p :}}$ It is false that all triangles are equilateral triangles.

(b) q:9 is a multiple of 4

Ans: We know that, Negation of the statement changes the truth value ${\text{T to F}}$ and ${\text{F to T}}{\text{.}}$

The negation of the statement is given by,

${\text{~ q : 9}}$ is not a multiple of ${\text{4}}{\text{.}}$

(c) r:A triangle has four sides.

Ans: We know that, Negation of the statement changes the truth value ${\text{T to F}}$ and ${\text{F to T}}{\text{.}}$

The negation of the statement is given by,

${\text{~ r :}}$ It is false that the triangle has four sides.

25. Write the negation of the following statements:

(i) Suresh lives in Bhopal or he lives in Mumbai.

Ans: We know that, Negation of the statement reverses the truth value ${\text{T to F}}$ and ${\text{F to T}}{\text{.}}$

The given statement is logically disjunction ${\text{p }} \vee {\text{ q}}{\text{.}}$

Therefore, the negation is given by,

${\text{~ (p }} \vee {\text{ q) :}}$ “Suresh does not live in Bhopal and he does not live in Mumbai”.

(ii) $x  +  y  =  y  +  x$ and $29$ is a prime number.

Ans: We know that, negation of the statement reverses the truth value ${\text{T to F}}$and ${\text{F to T}}{\text{.}}$

The given statement is conjunction ${\text{p }} \wedge {\text{ q}}{\text{.}}$

Therefore, the negation is given by,

${\text{~ (p }} \wedge {\text{ q) : x  +  y }} \ne {\text{ y  +  x and 29}}$is not a prime number.

26. Rewrite each of the following statements in the form of conditional statements:

(i) Mohan will be a good student if he studies hard.

Ans: We know that, a conditional statement is the logical statement obtained by using logical connector that is ${\text{p }} \to {\text{ q}}{\text{.}}$

The statement is of the form “${\text{q if p}}$“. 

Therefore, the conditional statement “${\text{if p then  q}}$” is given by,

${\text{p }} \to {\text{ q :}}$ “If Mohan studies hard, then he will be a good student”.

(ii) Ramesh will get dessert only if he eats his dinner.

Ans: We know that, A conditional statement is the logical statement obtained by using logical connector that is ${\text{p }} \to {\text{ q}}{\text{.}}$

The statement is of the form “${\text{p only if q}}$“. 

Therefore, the conditional statement “${\text{if p then q}}$“ is given by,

${\text{p }} \to {\text{ q :}}$“If Ramesh eats his dinner, then he will get dessert”.

(iii) When you sing, my ears hurt.

Ans: We know that, A conditional statement is the logical statement obtained by using logical connector that is ${\text{p }} \to {\text{ q}}{\text{.}}$

The conditional statement “${\text{if p then q}}$” is given by,

${\text{p }} \to {\text{ q :}}$ “If you sing, then my ears hurt”.

(iv) A necessary condition for the Indian team to win a cricket match is that the selection committee selects an all-rounder.

Ans: We know that, a conditional statement is the logical statement obtained by using logical connector that is ${\text{p }} \to {\text{ q}}{\text{.}}$

The statement is of the form “${\text{q is necessary for p}}$“. 

Therefore, the conditional statement “${\text{if p then q}}$” is given by,

${\text{p }} \to {\text{ q :}}$“If the team wins a cricket match then the selection committee selects an all-rounder”.

(v) A sufficient condition for Tara to visit New Delhi is that she goes to the Rashtrapati Bhawan.

Ans: We know that, a conditional statement is the logical statement obtained by using logical connector that is ${\text{p }} \to {\text{ q}}{\text{.}}$

The statement is of the form “${\text{p is sufficient for q}}$”. 

Therefore, the conditional statement “${\text{if p then q}}$“ is given by,

${\text{p }} \to {\text{ q :}}$ “If Tara goes to Rashtrapati Bhawan, then she visits New Delhi”.

27. Express in English, the statement $p \to  q$, where ${\text{p :}}$ It is raining today.

$q : 2  +  3  >  4.$

Ans: Given: The statement,

${\text{p :}}$ it is raining today

${\text{q : 2  +  3  >  4}}$

We know that, the conditional statement ${\text{p }} \to {\text{ q}}$ is expressed as if ${\text{p, then q}}{\text{.}}$

The required conditional statement ${\text{p }} \to {\text{ q}}$ can be expressed as,

${\text{p }} \to {\text{ q :}}$“If it is raining today, then ${\text{2  +  3  >  4}}$“.

28. Translate the following statements in symbolic form:

Ans: We know that symbolic forms use logical connectors to represent the statement logically.

The component statements can be written as,

${\text{p : If x  =  7 and y  =  4 and,}}$

${\text{q : x  +  y  =  11}}{\text{.}}$

Therefore, the symbolic form for the statement is ${\text{p }} \to {\text{ q}}{\text{.}}$

29. Form the bi-conditional of the following statements:

p: Today is 14th of August.

q: Tomorrow is Independence day.

Ans: We know that, Bi-conditional statement is the logical statement which uses if and only if as connector.

The bi-conditional statement is given by,

${\text{p }} \leftrightarrow {\text{ q :}}$ “Today is $14{\text{th}}$ of August if and only if tomorrow is Independence Day”.

30. Translate the following bi-conditional into symbolic form:

“$ABC$ is an equilateral triangle if and only if its each interior angle is $6{0^0}$”.

Ans: Given: Bi-conditional statement.

We know thatSymbolic forms use logical connectors to represent the statement logically.

The sub-statements can be represented as,

${\text{p : ABC}}$ is an equilateral triangle.

${\text{q :}}$ Each interior angle of triangle ${\text{ABC is 6}}{{\text{0}}^0}.$

Therefore, the bi-conditional statement is given by ${\text{p }} \leftrightarrow {\text{ q}}{\text{.}}$

31. Identify the quantifiers and write the negation of the following statements:

(i) There exists a number which is equal to its square.

Ans: We know that, Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

Quantifier in the statement is ‘There exists’.

Negation of the statement is given by, “There does not exist a number which is equal to its square”.

(ii) For all even integers $x,  {x^2}$ is also even.

Ans: We know that, Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘for all’ many more.

Quantifier in the statement is ‘For all’.

Negation of the statement is given by, “There exists an even integer ${\text{x such that }}{{\text{x}}^2}$ is not even”.

(iii) There exists a number which is a multiple of $6 \text{ and }9.$

Ans: We know that, Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

Quantifier in the statement is ‘There exists’.

Negation of the statement is given by, “There does not exist a number which is a multiple of both ${\text{6 and 9}}$”.

32. Show that the following statement is true.

$p :$ For any real numbers x, y if x  =  y, then  $2x  +  a  =  2y  +  a$ when $a \in  Z.$

Ans: We know that, use direct method and contrapositive method to show the given statement is true.

Using Direct Method, for two real numbers, ${\text{p  }} \Rightarrow {\text{ q}}$

${\text{x  =  y (given)}}$

$\Rightarrow {{ 2x  =  2y}}$

$\Rightarrow {{ 2x  +  a  =  2y  +  a, a }} \in {\text{ Z}}{\text{.}}$

By contrapositive Method, ${{q }} \Rightarrow {\text{ p}}$

$\Rightarrow {{ 2x  +  a }} \ne {{ 2y  +  a, a }} \in {\text{ Z}}$

$\Rightarrow {{ 2x }} \ne {{ 2y}}$

$\Rightarrow {{ x }} \ne {{ y}}$

33. Check the validity of the statements

(i) $r : 100$ is a multiple of $4 \text{ and } 5.$

Ans: We know that, 

Check \[{\text{(p }} \wedge {\text{ q) or (p }} \vee {\text{ q)}}{\text{.}}\] Then, verify the truth value.

Let \[{\text{r : p }} \wedge {\text{ q}}{\text{.}}\]

${\text{p : 100 is a multiple of 4}}{\text{.}}$

${\text{q : 100 is a multiple of 5}}{\text{.}}$

${\text{p is true and q is true}}{\text{.}}$

$\therefore {\text{ r is true and valid}}{\text{.}}$

(ii) $s : 60$ is a multiple of 3 or 5.

Ans: We know that, 

Check \[{\text{(p }} \wedge {\text{ q) or (p }} \vee {\text{ q)}}{\text{.}}\] Then, verify the truth value.

Let \[{\text{s : p }} \vee {\text{ q}}{\text{.}}\]

  p : 60 is a multiple of 3. 

  q : 60 is a multiple of 5. 

  p is true and q is true. 

  s is true and valid. 

Objective Type Question:

Choose the correct answer out of the four options given against each of the Examples \[16 to 18\] (M.C.Q.).

34. Which of the following is a statement?

(A) Roses are black.

(B) Mind your own business.

(C) Be punctual.

(D) Do not tell lies.

Ans: We know that, statement is a sentence which is either true or false.

Since, Roses are black is a false sentence. It is a statement.

All other sentences are neither true nor false. They are advice not statements.

Correct Option: A

35 The negation of the statement

“It is raining and weather is cold.” is

(A) It is not raining and weather is cold.

(B) It is raining or weather is not cold.

(C) It is not raining or weather is not cold.

(D) It is not raining and weather is not cold.

Ans: We know that, Negation of the statement reverses the truth value ${\text{T to F}}$ and ${\text{F to T}}{\text{.}}$

Let ${\text{p :}}$ It is raining, and

.${\text{q :}}$. Weather is cold.

The negation of the statement or conjunction ${\text{p }} \wedge {\text{ q}}$ is given by

${\text{~ (p }} \wedge {\text{ q) :}}$ “It is not raining or weather is not cold”.

Correct Option: C

36. Which of the following is the converse of the statement?

“If Billu secures good marks, then he will get a bicycle.”

(A) If Billu will not get a bicycle, then he will not secure good marks.

(B) If Billu will get a bicycle, then he will secure good marks.

(C) If Billu will get a bicycle, then he will not secure good marks.

(D) If Billu will not get a bicycle, then he will secure good marks.

Ans: We know that, Converse of the statement ${\text{p }} \to {\text{ q}}$ is given by ${\text{q }} \to {\text{ p}}{\text{.}}$

Since, ${\text{q }} \to {\text{ p}}$ is the converse of the statement ${\text{p }} \to {\text{ q}}{\text{.}}$

Therefore, the converse of the statement is “If Billu will get a bicycle, then he will secure good marks”.

Correct Option: B

Exercise:

1. Which of the following sentences are statements? Justify

(i) A triangle has three sides.

Ans: We know that, A statement is said to be a sentence which is either true or false but not both simultaneously. A triangle has three sides. It is a true statement.

(ii) 0 is a complex number.

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. \[0\] is a real complex number. It is a true statement.

(iii) Sky is red.

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. Sky is red. It is a false statement.

(iv) Every set is an infinite set.

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. Every set is a finite set. It is a false statement.

(v) 15 + 8 > 23

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. \[15{\text{  +  8  >  23}}{\text{.}}\] It is a false statement.

(vi) y + 9 = 7

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. \[{\text{y  +  9  =  7}}{\text{.}}\] Value of \[{\text{y}}\] is not given. It is not a statement.

(vii) Where is your bag?

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. Where is your bag? It is a question not a statement.

(viii) Every square is a rectangle.

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. Every square is a rectangle. It is a true statement.

(ix) Sum of opposite angles of a cyclic quadrilateral is \[18{0^0}.\]

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. Sum of opposite angles of a cyclic quadrilateral is \[{180^0}.\] It is a true statement.

(x) \[si{n^2}x  +  co{s^2}x  =  0.\]

Ans: We know that, a statement is said to be a sentence which is either true or false but not both simultaneously. \[{\sin ^2}{\text{x  +  co}}{{\text{s}}^2}{\text{x  =  0}}{\text{.}}\]It is a false statement.

2. Find the component statements of the following compound statements.

(i) Number \[7\] is prime and odd.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p :}}\] Number \[7\] is prime.

\[{\text{q :}}\] Number \[7\] is odd.

(ii) Chennai is in India and is the capital of Tamil Nadu.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p :}}\]Chennai is in India.

\[{\text{q :}}\] Chennai is the capital of Tamil Nadu.

(iii) The number 100 is divisible by 3, 11 and 5.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p : 100}}\] is divisible by \[3.\]

\[{\text{q : 100}}\] is divisible by \[11.\]

\[{\text{r : 100}}\] is divisible by \[5.\]

(iv) Chandigarh is the capital of Haryana and UP.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p :}}\]Chandigarh is the capital of Haryana.

\[{\text{q :}}\]Chandigarh is the capital of UP.

(v)\[\sqrt 7 \] is a rational number or an irrational number.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p : }}\sqrt 7 \] is a rational number.

\[{\text{q : }}\sqrt 7 \] is an irrational number.

(vi) 0 is less than every positive integer and every negative integer.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p : 0}}\] is less than every positive integer.

\[{\text{q : 0}}\] is less than every negative integer.

(vii) Plants use sunlight, water and carbon dioxide for photosynthesis.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p :}}\] Plants use sunlight for photosynthesis.

\[{\text{q : }}\] Plants use water for photosynthesis.

(viii) Two lines in a plane either intersect at one point or they are parallel.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p :}}\] Two lines in a plane intersect at one point.

\[{\text{q :}}\] Two lines in a plane are parallel.

(ix) A rectangle is a quadrilateral or a \[5\]sided polygon.

Ans: Component statement connects true and false statements with ‘and’.

\[{\text{p :}}\] A rectangle is a quadrilateral.

\[{\text{q :}}\] A rectangle is \[5{\text{-}}\]sided polygon.

3. Write the component statements of the following compound statements and check whether the compound statement is true or false.

(i) \[57\] is  divisible by 2 or 3.

Ans: First check the correctness of the statement. Then, write a component statement.

The compound statement is of form \['{\text{p }} \vee {\text{ q'}}{\text{.}}\] The truth value is \[{\text{T}}\] whenever either \[{\text{p or q}}\] both have truth value \[{\text{T}}{\text{.}}\]

The statement is true.

Component statements are

\[{\text{p : 57}}\] is divisible by \[{\text{2}}{\text{.}}\]

\[{\text{p : 57}}\] is divisible by \[3.\]

(ii) \[24\] is a multiple of 4 and 6.

Ans: First check the correctness of the statement. Then, write a component statement.

The compound statement is of form \['{\text{p }} \wedge {\text{ q'}}{\text{.}}\] The truth value is \[{\text{T}}\] whenever either \[{\text{p and q}}\] both have truth value \[{\text{T}}{\text{.}}\]

The statement is true.

Component statements are

\[{\text{p : 24}}\] is multiple of \[4.\]

\[{\text{q : 24}}\] is multiple of \[6.\]

(iii) All living things have two legs and two eyes.

Ans: First check the correctness of the statement. Then, write a component statement.

It is a false statement. As truth value is \[{\text{F}}\] whenever either \[{\text{p or q}}\] both have truth value \[{\text{F}}{\text{.}}\]

Component statements are

\[{\text{p :}}\]. All living things have two eyes.

\[{\text{q :}}\] All living things have two legs.

(iv) 2 is an even number and a prime number.

Ans: First check the correctness of the statement. Then, write a component statement.

It is a true statement.

Component statements are

\[{\text{p : 2}}\] is an even number.

\[{\text{q : 2}}\] is a prime number.

4. Write the negative on the following simple statements.

(i) The number 17 is prime.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement that the number \[17\] is prime will be the number \[17\] is not prime.

(ii) 2 + 7 = 6

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’. The negative statement of simple statement \[2{\text{  +  7  =  6}}\] will be \[2{\text{  +  7 }} \ne {\text{ 6}}{\text{.}}\]

(iii) Violets are blue.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement Violets are blue will be Violets are not blue.

(iv) \[\sqrt 5 \] is a rational number.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement \[\sqrt 5 \] is a rational number will be \[\sqrt 5 \] is not a rational number.

(v) 2 is not a prime number.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement \[2\] is not a prime number will be  \[2\] is not  a prime number.

(vi) Every real number is an irrational number.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement Every real number is an irrational number will be Every real number is not an irrational number.

(vii) Cow has four legs.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement Cow has four legs will be Cow has not four legs.

(viii) A leap year has 366 days.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement A leap year has \[366\]days will be A leap year has not \[366\]days.

(ix) All similar triangles are congruent.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement all similar triangles are congruent will be There exist similar triangles which are not congruent.

(x) Area of the circle is the same as the perimeter of the circles.

Ans: Negative statements are the statements which can be formed by connecting the word ‘Not’ with simple statements. The negative statements of the simple statements with word ‘Not’ will be formed by replacing ‘not’.

The negative statement of the simple statement Area of circle is same as the perimeter of the circle will be Area of circle is not same as the perimeter of the circle.

5. Translate the following statements into symbolic form:

(i) Rahul passed in Hindi and English.

Ans: First, break the statements in two parts. Then, use symbolic forms like conditional and bi-conditional. Symbolic forms are obtained using logical connections in statements. Here, the statement Rahul passed in Hindi and English. Therefore, the symbolic form of statement is,

\[{\text{p :}}\] Rahul passed in Hindi.

\[{\text{q :}}\] Rahul passed in English.

\[{\text{p }} \wedge {\text{ q :}}\] Rahul passed in Hindi and English.

(ii) x and y are even integers.

Ans: First, break the statements in two parts. Then, use symbolic forms like conditional and bi-conditional. Symbolic forms are obtained using logical connections in statements. Here, the statement is \[{\text{x and y}}\] are even integers. Therefore, the translation of the statement in symbolic form is,

\[{\text{p : x}}\] is even integers.

\[{\text{q : y}}\] is even integers.

\[{\text{p }} \cap {\text{ q : x and y}}\] are even integers.

(iii) 2, 3 and 6 are factors of 12

Ans: First, break the statements in two parts. Then, use symbolic forms like conditional and bi-conditional. Symbolic forms are obtained using logical connections in statements. Here, the statement is \[{\text{2, 3 and 6}}\] are factors of \[12.\] These will split into three symbolic forms. Therefore, the translation of the statement in symbolic form is,

\[{\text{p : 2}}\] is a factor of \[12.\]

\[{\text{q : 3}}\] is a factor of \[12.\]

\[{\text{r : 6}}\] is a factor of \[12.\]

\[{\text{p }} \wedge {\text{ q }} \wedge {\text{ r : 2, 3 and 6}}\] are factor of \[12.\]

(iv) Either \[x \text{ or }x  +  1\] is an odd integer.

Ans: First, break the statements in two parts. Then, use symbolic forms like conditional and bi-conditional. Symbolic forms are obtained using logical connections in statements. Here, the statement is either \[{\text{x or x  +  1}}\] is an odd integer. Therefore, the translation of statement in symbolic form is,

\[{\text{p : x}}\] is an odd integer.

\[{\text{q : x  +  1}}\] is an odd integer.

\[{\text{p }} \vee {\text{ q : Either x or (x  +  1)}}\] is an odd integer.

(v) A number is either divisible by 2 or  3.

Ans: First, break the statements in two parts. Then, use symbolic forms like conditional and bi-conditional. Symbolic forms are obtained using logical connections in statements. Here, the statement is a number is either divisible by \[{\text{2 or 3}}{\text{.}}\] Therefore, the translation in symbolic form is,

\[{\text{p :}}\] A number is divisible by \[2.\]

\[{\text{q :}}\] A number is divisible by \[3.\]

\[{\text{p }} \vee {\text{ q :}}\] A number is either divisible by \[{\text{2 or 3}}{\text{.}}\]

(vi) Either x  =  2 or x  =  3 is a root of \[3{x^2}  -  x  -  10  =  0.\]

Ans: First, break the statements in two parts. Then, use symbolic forms like conditional and bi-conditional. Symbolic forms are obtained using logical connections in statements. Here, the statement is either \[{\text{x  =  2 or x  =  3}}\] is a root of \[{\text{3}}{{\text{x}}^2}{\text{  -  x  -  10   =  0}}{\text{.}}\]Therefore, the translation in symbolic form is,

\[{\text{p : x  =  2}}\] is a root of \[{\text{3}}{{\text{x}}^2}{\text{  -  x  -  10  =  0}}{\text{.}}\]

\[{\text{q : x  =  3}}\] is a root of \[{\text{3}}{{\text{x}}^2}{\text{  -  x  -  10  =  0}}{\text{.}}\]

\[{\text{p }} \vee {\text{ q : Either x  =  2 or x  =  3}}\] is a root of \[{\text{3}}{{\text{x}}^2}{\text{  -  x  -  10  =  0}}{\text{.}}\]

(vii) Students can take Hindi or English as an optional paper.

Ans: First, break the statements in two parts. Then, use symbolic forms like conditional and bi-conditional. Symbolic forms are obtained using logical connections in statements. Here, the statement is students can take Hindi or English as an optional paper. Therefore, the translation in symbolic form is,

\[{\text{p :}}\] students can take Hindi as an optional paper.

\[{\text{q :}}\] students can take Hindi as an optional paper.

\[{\text{p }} \vee {\text{ q :}}\] students can take Hindi or English as an optional paper.

6. Write down the negation of the following compound statements.

(i) All rational numbers are real and complex.

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p :}}\] All rational numbers are real.

\[{\text{q :}}\] All rational numbers are complex.

\[{\text{~ p :}}\] All rational numbers are not real.

\[{\text{~}}\;{\text{q :}}\] All rational numbers are complex.

Now, use the properties of negations to obtain the

Therefore, the negation of the above statement is given by,

\[{\text{~ (p }} \wedge {\text{ q) :}}\] All rational numbers are not real or not complex.

(ii) All real numbers are rationals or irrationals.

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p :}}\] All real numbers are rational.

\[{\text{q :}}\] All real numbers are irrational

Now, taking the complete negation of above statements.

Therefore, the negation of the above statement is given by,

\[{\text{~ (p }} \vee {\text{ q) :}}\] All real numbers are not rational and all real numbers are not irrational.

(iii) \[x  =  2 and x  =  3\] are roots of the Quadratic equation \[{x^2}  -  5x  +  6  =  0.\].

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p : x  =  2}}\] is root of quadratic equation \[{{\text{x}}^2}{\text{  -  5x  +  6  =  0}}{\text{.}}\]

\[{\text{q : x  =  3}}\] is root of quadratic equation \[{{\text{x}}^2}{\text{  -  5x  +  6  =  0}}{\text{.}}\]

Now, taking the complete negation of above statements.

Therefore, the negation of the above statement is given by,

\[{\text{~ (p }} \wedge {\text{ q) : x  =  2}}\] is not a root of quadratic equation \[{{\text{x}}^2}{\text{  -  5x  +  6  =  0 or x  =  3}}\] is not a root of the quadratic equation \[{{\text{x}}^2}{\text{  -  5x  +  6  =  0}}{\text{.}}\]

(iv) A triangle has either 3-sides or 4- sides.

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p :}}\]A triangle has \[3\; - \]sides.

\[{\text{q :}}\]A triangle has \[4\; - \]sides.

Now, taking the complete negation of above statements

Therefore, the negation of the above statement is given by,

\[{\text{~ (p }} \vee {\text{ q) :}}\]A triangle has neither \[{\text{3  -  sides nor 4  -  sides}}{\text{.}}\]

(v) 35 is a prime number or a composite number.

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p : 35}}\] is a prime number.

\[{\text{q : 35}}\] is a composite number.

Now, taking the complete negation of above statements.

Therefore, the negation of the above statement is given by,

\[{\text{~ (p }} \vee {\text{ q) : 35}}\] is not a prime number  and it is not a composite number.

(vi) All prime integers are either even or odd.

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p :}}\] All prime integers are even.

\[{\text{q :}}\] All prime integers are odd.

Now, taking the complete negation of above statements.

Therefore, the negation of the above statement is given by,

\[{\text{~ (p }} \vee {\text{ q) :}}\] All prime integers are not even and all prime integers are not odd.

(vii) \[| x |\] is equal to either \[x or  -  x.\]

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p : | x |}}\] is equal to \[{\text{x}}{\text{.}}\]

\[{\text{q : | x |}}\] is equal to \[{\text{ -  x}}{\text{.}}\]

Now, taking the complete negation of above statements.

Therefore, the negation of the above statement is given by

\[{\text{~ (p }} \vee {\text{ q) : | x |}}\] is not equal to \[{\text{x}}\] and it is not equal to \[{\text{ -  x}}{\text{.}}\]

(viii) 6 is divisible by 2 and 3.

Ans: Use the following identities to obtain the negation of the compound statements.

${\text{(i) ~ (p }} \wedge {\text{ q)  =  ~ p }} \vee {\text{ ~ q}}$

${\text{(ii) ~ (p }} \vee {\text{ q)  =  ~ p }} \wedge {\text{ ~ q}}$

First, split the given compound statements. Then, taking negation of individual statements.

Applying the operations step by step, we get

\[{\text{p : 6}}\] is divisible by \[2.\]

\[{\text{q : 6}}\] is divisible by \[{\text{3}}{\text{.}}\]

Now, taking the complete negation of above statements.

Therefore, the negation of the above statement is given by,

\[{\text{~ (p }} \wedge {\text{ q) : 6}}\] is not divisible by \[{\text{2}}\] or it is not divisible by \[{\text{3}}{\text{.}}\]

7. Rewrite each of the following statements in the form of conditional statements.

(i) The square of an odd number is odd.

Ans: Use the common expression of conditional statement \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The conditional statement is given by If the number is an odd number, then its square is an odd number.

(ii) You will get a sweet fish after the dinner.

Ans: Use the common expression of conditional statement \[{\text{p }} \to {{ q}}{\text{.}}\]

The conditional statement is given by If you take the dinner, then you will get a sweet dish.

(iii) You will fail, if you will not study.

Ans: Use the common expression of conditional statement \[{{p }} \to {{ q}}{\text{.}}\]

The conditional statement is given by If you will not study, then you will fail.

(iv) The unit digit of an integer is \[0 \text{ or } 5,\] if it is divisible by \[{{5}}{{.}}\]

Ans: Use the common expression of conditional statement \[{{p }} \to {\text{ q}}{\text{.}}\]

The conditional statement is given by If an integer is divisible by \[5,\] then its unit digits are \[{\text{0 and 5}}{\text{.}}\]

(v) The square of a prime number is not prime.

Ans: Use the common expression of conditional statement \[{{p }} \to {\text{ q}}{\text{.}}\]

The conditional statement is given by If the number is prime, then its square is not prime.

(vi) \[2b  =  a  +  c,\] if a, b and c are in AP.

Ans: Use the common expression of conditional statement \[{{p }} \to {\text{ q}}{\text{.}}\]

The conditional statement is given by If \[{\text{a, b and c}}\] are in AP, then \[{\text{2b  =  a  +  c}}{\text{.}}\]

8. Form the bi-conditional statement \[p \ll  q,\] where

(i) p: The unit digits of an integer is zero.

q: It is divisible by \[{\text{5}}{\text{.}}\]

Ans: A bi-conditional statement is a statement which is true if and only if both the conditional statements are true. First, combine the conditional statements. Then, write their converse in if and only if form.

The bi-conditional statement is given by \[{\text{p }} \leftrightarrow {\text{ q :}}\] The unit digit of an integer is zero, if and only if it is divisible by \[{\text{5}}{\text{.}}\]

(ii) p: A natural number n is odd.

q: Natural number n is not divisible by 2

Ans: A bi-conditional statement is a statement which is true if and only if both the conditional statements are true. First, combine the conditional statements. Then, write their converse in if and only if form.

The bi-conditional statement is given by \[{\text{p }} \leftrightarrow {\text{ q :}}\]A natural number \[{\text{n}}\] is odd if and only if it is not divisible by \[2.\]

(iii) p: A triangle is an equilateral triangle.

q: All three sides of a triangle are equal.

Ans: A bi-conditional statement is a statement which is true if and only if both the conditional statements are true. First, combine the conditional statements. Then, write their converse in if and only if form.

The bi-conditional statement is given by \[{\text{p }} \leftrightarrow {\text{ q :}}\]A triangle is an equilateral triangle if and only if all three sides of the triangle are equal.

9. Write down the contrapositive of the following statements.

(i) If x  =  y and y  =  3, then x  =  3.

Ans: As the statement \[{\text{(~ q) }} \to {\text{ (~ p) }}\] is called contrapositive of the statement  \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The contrapositive of the statement is given by If \[{\text{x }} \ne {\text{ 3, then x }} \ne {\text{ y or y }} \ne {\text{ 3}}{\text{.}}\]

(ii) If \[{\text{n}}\] is a natural number, then \[{\text{n}}\] is an integer.

Ans: As the statement \[{\text{(~ q) }} \to {\text{ (~ p) }}\] is called contrapositive of the statement  \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The contrapositive of the statement is given by If \[{\text{n}}\] is not an integer, then it is not a natural number.

(iii) If all three sides of a triangle are equal, then the triangle is equilateral.

Ans: As the statement \[{\text{(~ q) }} \to {\text{ (~ p) }}\] is called contrapositive of the statement  \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The contrapositive of the statement is given by If the triangle is not equilateral, then all three sides of the triangle are not equal.

(iv) If \[x \text{ and } y\] are negative integers, then \[xy\] is positive.

Ans: As the statement \[{\text{(~ q) }} \to {\text{ (~ p) }}\] is called contrapositive of the statement  \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The contrapositive of the statement is given by If \[{\text{xy}}\] is not a positive integer, then either \[{\text{x or y}}\] is not a negative integer.

(v) If the natural number \[{\text{n}}\] is divisible by 6, then n is divisible by 2 and 3.

Ans: As the statement \[{\text{(~ q) }} \to {\text{ (~ p) }}\] is called contrapositive of the statement  \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The contrapositive of the statement is given by If natural number \[{\text{n}}\] is not divisible by \[2{\text{ or 3, then n is not divisible by 6}}{\text{.}}\]

(vi) If it snows, then the weather will be cold.

Ans: As the statement \[{\text{(~ q) }} \to {\text{ (~ p) }}\] is called contrapositive of the statement  \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The contrapositive of the statement is given by The weather will not be cold, if it does not snow.

(vii) If \[{\text{x}}\] is a real number such that  \[0  <  x  <  1, \text{ then } {x^2}  <  1.\]

Ans: As the statement \[{\text{(~ q) }} \to {\text{ (~ p) }}\] is called contrapositive of the statement  \[{\text{p }} \to {\text{ q}}{\text{.}}\]

The contrapositive of the statement is given by If \[{{\text{x}}^2}{\text{ not  <  1, then x}}\] is not a real number such that \[0{\text{  <  x  <  1}}{\text{.}}\]

10. Write down the converse of the following statements.

(i) If a rectangle \['{\text{R'}}\] is a square, then \[{\text{R}}\] is a rhombus.

Ans: Converse of the statement is given by

The converse of the following statement is if the rectangle \[{\text{'R'}}\] is rhombus, then it is a square.

(ii) If today is Monday, then tomorrow is Tuesday.

Ans: Converse of the statement is given by

The converse of the following statement is if tomorrow is Tuesday, then today is Monday.

(iii) If you go to Agra, then you must visit the Taj Mahal.

Ans: Converse of the statement is given by

The converse of the following statement is if you must visit Taj Mahal , you go to Agra.

(iv) If the sum of squares of two sides of a triangle is equal to the square of the third side of a triangle, then the triangle is right angled.

Ans: Converse of the statement is given by

The converse of the following statement is if the triangle is right angle, then the sum of squares of two sides of a triangle is equal to the square of the third side.

(v) If all three angles of a triangle are equal, then the triangle is equilateral.

Ans: Converse of the statement is given by

The converse of the following statement is if the triangle is equilateral, then all three angles of the triangle are equal.

(vi) If \[x : y  =  3 : 2, \text{ then } 2x  =  3y.\]

Ans: Converse of the statement is given by

The converse of the following statement is if \[{\text{2x  =  3y, then x : y  =  3 : 2}}{\text{.}}\]

(vii) If \[{\text{S}}\] is a cyclic quadrilateral, then the opposite angles of \[{\text{S}}\] are supplementary.

Ans: Converse of the statement is given by

The converse of the following statement is if the opposite angles of a quadrilateral are supplementary, then \[{\text{S}}\] is cyclic.

(viii) If \[{\text{x}}\] is zero, then \[{\text{x}}\] is neither positive nor negative.

Ans: Converse of the statement is given by

The converse of the following statement is if \[{\text{x}}\] is neither positive nor negative, then \[{\text{x is 0}}{\text{.}}\]

(ix) If two triangles are similar, then the ratio of their opposite sides are equal.

Ans: Converse of the statement is given by

The converse of the following statement is if the ratio of corresponding sides of two triangles are equal, then triangles are similar.

11. Identify the quantifiers in the following statements.

(i) There exists a triangle which is not equilateral.

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

There exists

(ii) For all real numbers \[x \text{ and } y, xy  =  yx.\]

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

For all

(iii) There exists a real number which is not a rational number.

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

There exists

(iv) For every natural number \[x, x  +  1\] is also a natural number.

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

For every

(v) For all real numbers \[x \text{ with } x  >  3, {x^2}\] is greater than \[9.\]

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

For all

(vi). There exists a triangle which is not an isosceles triangle.

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

There exists

(vii). For all negative integers \[x, {x^3}\] is also a negative integer.

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

 For all

(viii) There exists a statement in the above statements which is not true.

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

There exists

(ix) There exists a even prime number other than \[2.\]

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

There exists

(x) There exists a real number \[{\text{x}}\] such that \[{x^2}  +  1  =  0.\]

Ans: Quantifiers are phrases like ‘There exist’ and ‘For every’, ‘For all’ many more.

There exists

12. Prove by direct method that for any integer \['n', {n^3}  -  n\] is always even.

Ans: Given: \[{\text{'n', }}{{\text{n}}^3}{\text{  -  n}}\] for any integer.

In direct method to show a statement, if\[{\text{p then q}}\] is true, assume \[{\text{p}}\] is true and show \[{\text{q}}\] is true.

${\text{Case 1 :  n is even,}}$

Let ${{ n  =  2K, K }} \in {{ N}}$

$\Rightarrow {{ }}{{{n}}^3}{{  -  n  =  (2K}}{{{)}}^3}{{  -  (2K)}}$

$\Rightarrow {n}^3-n=2K(4{K}^2-1)$

$\Rightarrow {{ }}{{{n}}^3}{{  -  n  =  2m, m  =  K(4}}{{{K}}^2}{{  -  1)}}. $

$\therefore {{ (}}{{{n}}^3}{{  -  n}}){\text{ is even}}{\text{.}}$

${\text{Case 2 : n is odd,}}$

Let ${{ n  =  2K  +  1, K }} \in {{ N}}$

$\Rightarrow {{ }}{{{n}}^3}{{  -  n  =  (2K  +  1}}{{{)}}^3}{{  -  (2K  +  1)}}$

$\Rightarrow {{ }}{{{n}}^3}{{  -  n  =  (2K  +  1)}}[{{{(2K  +  1)}}^2}{{  -  1]}}$

$\Rightarrow {{ }}{{{n}}^3}{{  -  n  =  (2K  +  1)(4}}{{{K}}^2}{{  +  4K)}}$

$\Rightarrow {{ }}{{{n}}^3}{{  -  n  =  4K(2K  +  1)(K  +  1)}}$

$\Rightarrow {{ }}{{{n}}^3}{{  -  n  =  2p, p  =  2K(2K  +  1)(K  +  1)}}$

${\text{So, }}{{{n}}^3}{\text{  -  n is even for odd n}}{\text{.}}$

$\therefore {\text{ }}{{{n}}^3}{\text{  -  n is always even}}{\text{.}}$

13. Check validity of the following statement

(i) \[p : 125\] is divisible by 5 and 7.

Ans: Check \[{\text{(p }} \wedge {\text{ q) or (p }} \vee {\text{ q)}}{\text{.}}\] Then, verify the truth value.

Here,

\[{\text{p : 125}}\] is divisible by \[{\text{5 and 7}}{\text{.}}\]

\[{\text{Let a : 125}}\] is divisible by \[5.\]

\[{\text{b : 125}}\] is divisible by \[7.\]

${\text{a is true, b is false}}{\text{.}}$

$\Rightarrow {\text{ a }} \wedge {\text{ b is false}}{\text{.}}$

$\therefore {\text{ p is not valid}}{\text{.}}$

(ii) \[q : 132\] is a multiple of \[3 \text{ or } 11.\]

Ans: Check \[{\text{(p }} \wedge {\text{ q) or (p }} \vee {\text{ q)}}{\text{.}}\] Then, verify the truth value.

Here,

\[{\text{q : 131}}\] is a multiple of \[3{\text{ or 11}}{\text{.}}\]

\[{\text{Let a : 131}}\] is multiple of \[3.\]

\[{\text{b : 131}}\] is a multiple of \[11.\]

${\text{a is true, b is false}}{\text{.}}$

$\Rightarrow {\text{ a }} \vee {\text{ b is true}}{\text{.}}$

$\therefore {\text{ q is valid}}{\text{.}}$

14. Prove the following statement by contradiction method

\[p :\] The sum of an irrational number and a rational number is irrational.

Ans: Use a contradiction method. Assume the statement \[{\text{p}}\] either true or false.

Let the statement \[{\text{p}}\] be false.

Now, let \[\sqrt {\text{a}}\] be irrational and \[{\text{b}}\] is a rational number.

$\Rightarrow {\text{ }}\sqrt {\text{a}} {\text{  +  b  =   r}}$

$\Rightarrow {\text{ }}\sqrt {\text{a}} {\text{  =  r  -   b}}$

$\sqrt {\text{a}} {\text{ is irrational, but (r  -   b) is rational}}{\text{.}}$

This is a contradiction. Our assumption is wrong.

Therefore, the statement \[{\text{p}}\] is true.

15. Prove by direct method that for any real number \[x, y \text{ if } x  =  y, \text{ then } {x^2}  =  {y^2}.\]

Ans: Given: \[{\text{x  =  y}}{\text{.}}\] Use a direct method, assume \[{\text{p}}\] is true and show \[{\text{q}}\] is true.

${\text{p : x  =  y, x, y }} \in {\text{ R}}$

${\text{On squaring, we get}}$

${{\text{x}}^2}{\text{  =  }}{{\text{y}}^2}{\text{ : q}}$

$\therefore {\text{ p }} \Rightarrow {\text{ q}}{\text{.}}$

16. Using contrapositive methods prove that, if \[{n^2}\] is an even integer, then \[{\text{n}}\] is also an even integer.

Ans: Given: \[{{\text{n}}^2}\] is an even integer.

Use the contrapositive method, assume \[{\text{~ q}}\] and show \[~{\text{ p}}\] is true.

Let \[{\text{p : }}{{\text{n}}^2}\] is an even integer.

\[{\text{q : n}}\] is also an even integer.

Now, assume \[~{\text{ q}}\] is true. Then, \[{{\text{n}}^2}\] is not an even integer.

$\Rightarrow {\text{ }}{{\text{n}}^2}{\text{ is not an even integer}}{\text{.}}$

$\Rightarrow {\text{~ p is true}}{\text{.}}$

Hence proved.

Objective Type Questions:

Choose the correct answer out of the four options given against each of the Exercises 17 to 36 (M.C.Q.).

17. Which of the following is a statement?

(A) \[{\text{x}}\] is a real number

(B) Switch off the fan

(C) 6 is a natural number

(D) Let me go

Ans: Use the definition of statement. Statement is a sentence which is either true or false.

Since, \[6\] is a natural number is true.

Therefore, it is a statement.

Correct Option: C

18. Which of the following is not a statement?

(A) Smoking is injurious to health

(B) \[2  +  2  =  4\]

(C) \[2\] is the only even prime number

(D) Come here

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

 ‘Come here’ is neither true nor false. It is an order.

Therefore, it is not a statement.

Correct Option: D

19. The connective in the statement 2  +  7  >  9 or  2  +  7  <  9 is

(A) And

(B) Or

(C) \[ > \]

(D) \[ < \]

Ans: Connective is a logical symbol or word which signifies particular instruction. Connectives are used to join two or more statements. The statement \['2  +  7  >  9 \text{ or } 2  +  7  <  9'\] is connected using or connective.

Therefore, connective is or.

Correct Option: B

20. The connective in the statement

“Earth revolves round the sun and Moon is satellite of earth” is

(A) Or

(B) Earth

(C) Sun

(D) And

Ans: Connective is a logical symbol or word which signifies particular instruction. Connectives are used to join two or more statements.

The statement “Earth revolves around the sun and Moon is a satellite of earth” is connected using and connective.

Therefore, connective is and.

Correct Option: D

21. The negation of the statement “A circle is an ellipse” is

(A) An ellipse is a circle

(B) An ellipse is not a circle

(C) A circle is not an ellipse

(D) A circle is an ellipse

Ans: Negation of the statement is obtained by connecting or removing the word ‘Not’ in the statement.

Let \[{\text{p :}}\] A circle is an ellipse.

Therefore, the negation of the statement is given by,

\[~p :\] A circle is not an ellipse.

Correct Option: C

22. The negation of the statement “\[7\] is greater than \[8\] “ is

(A) 7 is equal to 8

(B) 7 is not greater than 8

(C) 8 is less than 7

(D) None of these

Ans: Negation of the statement is obtained by connecting or removing the word ‘Not’ in the statement.

Let \[{\text{p :}}\] “\[{\text{7}}\] is greater than \[{\text{8}}\] “.

Therefore, the negation of the statement is given by,

\[~p :\] “\[{\text{7}}\] is not greater than \[{\text{8}}\] “.

Correct Option: B

23. The negation of the statement “\[72\] is divisible by 2 and 3” is

(A) \[72\] is not divisible by 2 or 72 is not divisible by \[3.\]

(B) \[72\] is not divisible by 2 and 72 is not divisible by \[3.\]

(C) \[72\] is divisible by 2 and 72 is not divisible by \[3.\]

(D) \[72\] is not divisible by 2 and 72 is divisible by \[3.\]

Ans: Negation of the statement is obtained by connecting or removing the word ‘Not’ in the statement.

Let \[{\text{p : 72}}\] is divisible by \[2{\text{ and 3}}{\text{.}}\]

\[{\text{q : 72}}\] is divisible by \[2.\]

\[{\text{r : }}72\] is divisible by \[3.\]

\[{\text{~ q : 72}}\] is not divisible by \[2.\]

\[{\text{~ r : }}72\] is not divisible by \[3.\]

${\text{~ (q }} \wedge {\text{ r) : ~ q }} \vee {\text{ ~ r}}$

$\Rightarrow {\text{ 72 is not divisible by 2 or 72 is not divisible by 3}}{\text{.}}$

Correct Option: B

24. The negation of the statement “Plants take in \[C{O_2}\] and give out \[{O_2}\] “ is

(A) Plants do not take in \[C{O_2}\] and do not given out \[{O_2}\]

(B) Plants do not take in \[C{O_2}\] and do not give out \[{O_2}\]

(C) Plants take in \[C{O_2}\] and do not give out \[{O_2}\]

(D) Plants take in \[C{O_2}\] or do not give out \[{O_2}\]

Ans: Negation of the statement is obtained by connecting or removing the word ‘Not’ in the statement.

Let us consider the statements as,

\[{\text{p :}}\] Plants take in \[{\text{C}}{{\text{O}}_2}\] and give out \[{{\text{O}}_2}\].

\[{\text{q :}}\] Plants take in \[{\text{C}}{{\text{O}}_2}\].

\[{\text{r :}}\] Plants give out \[{{\text{O}}_2}.\]

\[{\text{~ q :}}\] Plants do not take in \[{\text{C}}{{\text{O}}_2}.\]

\[{\text{~ r :}}\] Plants do not give out \[{{\text{O}}_2}.\]

Therefore, the negation of the statement is given by,

\[{\text{~ (q }} \wedge {\text{ r) :}}\] Plants do not take in \[{\text{C}}{{\text{O}}_2}\] or do not give out \[{{\text{O}}_2}.\]

Correct Option: B

25. The negative of the statement “Rajesh or Rajni lived in Bangalore” is

(A) Rajesh did not live in Bengaluru or Rajni lives in Bangalore

(B) Rajesh lives in Bengaluru and Rajni did not live in Bangalore

(C) Rajesh did not live in Bengaluru and Rajni did not live in Bangalore

Ans: Negation of the statement is obtained by connecting or removing the word ‘Not’ in the statement.

Let us consider the statements as,

\[{\text{p :}}\] Rajesh or Rajni lived in Bangalore.

\[{\text{q :}}\] Rajesh lived in Bangalore.

\[{\text{r :}}\] Rajni lived in Bangalore.

\[{\text{~ q :}}\] Rajesh did not live in Bangalore.

\[{\text{~ r :}}\] Rajni did not live in Bangalore.

Therefore, the negation of the statement is given by,

\[{\text{~ (q }} \vee {\text{ r) :}}\] Rajesh did not live in Bangalore and Rajni did not live in Bangalore.

Correct Option: C

26. The negation of the statement “ \[101\] is not a multiple of \[3\] “ is

\[(A) 101\] is  a multiple of \[3\]

\[(B) 101\] is a multiple of \[2\]

\[(C) 101\] is an odd number

\[(D) 101\] is an even number

Ans: Negation of the statement is obtained by connecting or removing the word ‘Not’ in the statement.

Let \[{\text{p : 101}}\]  is not a multiple of \[3.\]

Therefore, the negation of the statement is given by,

\[{\text{~ p : 101}}\] is a multiple of \[3.\]

Correct Option: A

27. The contrapositive of the statement

“If \[7\] is greater than 5, then 8 is greater than \[6\] “ is

(A) If \[8\] is greater than 6, then 7 is greater than \[5.\]

(B) If \[8\] is not greater than 6, then 7 is greater than \[5.\]

(C) If \[8\] is not greater than 6, then 7 is not greater than \[5.\]

(D) If \[8\] is greater than 6, then 7 is not greater than \[5.\]

Ans: Contrapositive of any statement is given by \[(~{\text{ q) }} \to {\text{ (~ p)}}{\text{.}}\]

Let us split the statement to obtain contrapositive of the statement,

\[{\text{p : 7 }}\] is greater than \[5.\]

\[{\text{q : 8}}\] is greater than \[6.\]

\[\therefore {\text{ p }} \to {\text{ q}}\]

Now, taking negation of the conditional statements,

\[~{\text{ p : 7}}\] is not greater than \[5.\]

\[{\text{~ q : 8}}\] is not greater than \[6.\]

Now, applying the condition of contrapositive of statement \[{\text{(~ q) }} \to {\text{ (~ p)}}{\text{.}}\]

Therefore, the contrapositive of the statement is given by,

\[{\text{(~ q) }} \to {\text{ (~ p) :}}\] If \[8\] is not greater than \[6,{\text{ then 7}}\] is not greater than \[5.\]

Correct Option: C

28. The converse of the statement “If \[x  >  y, \text{ then } x  +  a  >  y  +  a\]” is

(A) If \[x  <  y, \text{ then } x  +  a  <  y  +  a\]

(B) If \[x  +  a  >  y  +  a, \text{ then } x  >  y\]

(C) If \[x  <  y, \text{ then } x  +  a  >  y  +  a\]

(D) If \[x  >  y, \text{ then } x  +  a  <  y  +  a\]

Ans: Converse of statement is the logic in which the two statements are reversed. Converse of statement \[{\text{p }} \to {\text{ q is q }} \to {\text{ p}}{\text{.}}\]

${\text{Let p : x  >  y}}$

${\text{q : x  +  a  >  y  +  a}}$

${\text{p }} \to {\text{ q}}$

Therefore, the converse of the statement is given by,

${\text{q }} \to {\text{ p}}$

$\therefore {\text{ if x  +  a  >  y  +  a, then x  >  y}}{\text{.}}$

Correct Option: B

29. The converse of the statement “ If the sun is not shining, then sky is filled with clouds “ is

(A) If sky is filled with clouds, then the sun is not shining

(B) If sun is shining, then sky is filled with clouds

(C) If the sky is clear, then the sun is shining.

(D) If sun is not shining, then sky is not filled with clouds

Ans: Converse of statement is the logic in which the two statements are reversed. Converse of statement \[{\text{p }} \to {\text{ q is q }} \to {\text{ p}}{\text{.}}\]

Let us consider the statements as,

\[{\text{p :}}\] Sun is not shining.

\[{\text{q :}}\] Sky is filled with clouds.

Now, the converse of the statement \[{\text{p }} \to {\text{ q is q }} \to {\text{ p}}{\text{.}}\]

Therefore, the converse of the statement is given as If the sky is filled with clouds, then the sun is not shining.

Correct Option: A

30. The contrapositive of the statement “ If p, then q”, is

(A) if $q, \text{ then } p$

(B) if $p, \text{ then } ~q$

(C) if $~ q, \text{ then } ~p$

(D) if $~ p, \text{ then } ~q$

Ans: Contrapositive of any statement is given by \[(~{\text{ q) }} \to {\text{ (~ p)}}{\text{.}}\]

The logical statement \[{\text{p }} \to {\text{ q}}\] represents the statement “ If \[{\text{p, then q}}\]”.

The contrapositive of the logical statement \[{\text{p }} \to {\text{ q}}\] is given by \[(~{\text{ q) }} \to {\text{ (~ p)}}{\text{.}}\]

Therefore, the contrapositive of statement “ If \[{\text{p, then q}}\]” is given by \[{\text{if ~ q, then ~ p}}{\text{.}}\]

Correct Option: C

31. The statement “If \[{x^2}\] is not even, then \[x\] is not even” is converse of the statement

(A) If \[{x^2}\] is odd, then \[x\] is even

(B) If \[x\] is not even, then \[{x^2}\] is not even

(C) If \[x\] is even, then \[{x^2}\] is even

(D) If \[x\] is odd, then \[{x^2}\] is even

Ans: Converse of statement is the logic in which the two statements are reversed. Converse of statement \[{\text{p }} \to {\text{ q is q }} \to {\text{ p}}{\text{.}}\]

Let us consider the statements as,

\[{\text{p : }}{{\text{x}}^2}\] is not even.

\[{\text{q : x}}\] is not even.

Now, the converse of the statement \[{\text{p }} \to {\text{ q is q }} \to {\text{ p}}{\text{.}}\]

Therefore, the converse of the statement is given as If \[{\text{x}}\] is not even, then \[{{\text{x}}^2}\] is not even.

Correct Option: B

32. The contrapositive of the statement ‘If Chandigarh is capital of Punjab, then Chandigarh is in India’ is

(A) If Chandigarh is not in India, then Chandigarh is not the capital of Punjab

(B) If Chandigarh is in India, then Chandigarh is capital of Punjab

(C) If Chandigarh is not capital of Punjab, then Chandigarh is not capital of India

(D) If Chandigarh is capital of Punjab, then Chandigarh is not in India

Ans: Contrapositive of any statement is given by \[(~{\text{ q) }} \to {\text{ (~ p)}}{\text{.}}\]

Let us consider the statements as,

\[{\text{p :}}\]  Chandigarh is the capital of Punjab.

\[{\text{q :}}\]Chandigarh is in India.

Now, taking negation of the statements,

\[{\text{~ p :}}\] Chandigarh is not the capital of Punjab.

\[{\text{~ q :}}\] Chandigarh is not in India.

The contrapositive of the logical statement \[{\text{p }} \to {\text{ q}}\] is given by \[(~{\text{ q) }} \to {\text{ (~ p)}}{\text{.}}\]

Therefore, the contrapositive of the statement is given by if Chandigarh is not in India, then Chandigarh is not the capital of Punjab.

Correct Option: A

33. Which of the following is the conditional \[p  \to q\] ?

\[(A) q\] is sufficient for \[{\text{p}}\]

\[(B) p\] is necessary for \[{\text{q}}\]

\[(C) p\] only if \[{\text{q}}\]

 (D) If \[q, \text{ then } p\]

Ans: Conditional statements are formed using a logical connector between two statements.

The conditional statement \[{\text{p }} \to {\text{ q}}\] is similar to \['{\text{ p only if q '}}{\text{.}}\]

Correct Option: C

34. The negation of the statement “ The product of

(A) It is false that the product of 3 and 4 is 9.

(B) The product of 3 and 4 is 12.

(C) The product of 3 and 4 is not 12.

(D) It is false that the product of 3 and 4 is not 9.

Ans: Negation of the statement is obtained by changing truth value \[{T \to F}\] and \[F \to T\]

The negation of the given statement is “It is false that that the product of

Correct Option: A

35. Which of the following is not a negation of

“A natural number is greater than zero”.

(A) A natural number is greater than zero

(B) It is false that a natural number is greater than zero

(C) It is false that a natural number is not greater than zero

(D) None of the above

Ans: Negation of the statement is obtained by changing truth value \[T \to F\] and \[F \to T\]

The false negation of the given statement is given by “It is false that a natural number is not greater than zero”.

Correct Option: C

36. Which of the following statements is a conjunction?

(A) Ram and Shyam are friends.

(B) Both Ram and Shyam are tall.

(C) Both Ram and Shyam are enemies.

(D) None of the above.

Ans: Conjunction is obtained by using connector ‘AND’ between the two statements.

None of the statements is formed using connector ‘AND’ in the given options.

Therefore, none of the above given statements is in conjunction.

Correct Option: D

37. State whether the following sentences are statements are not:

(i) The angles opposite to equal sides of a triangle are equal.

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

The sentence the angles opposite to equal sides of a triangle are equal is true. Therefore, it is a statement.

(ii) The moon is a satellite of earth.

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

The sentence the moon is a satellite of earth is true. Therefore, it is a statement.

(iii) May God bless you!

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

Since it is an exclamation. It is not a statement.

(iv) Asia is a continent.

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

The sentence Asia is a continent is true. Therefore, it is a statement.

(v) How are you?

Ans: Use basic definitions of statements. Statement is a sentence which is either true or false.

Since, it is a question. It is not a statement.

Important Topics for Class 11 Maths Chapter 14 Mathematical Reasoning

The Important Topics for Chapter 14 are as follows:

• Definition of statements

• Simple statements

• Compound statements

• Basic logical connectives

• Definition and meaning of conjunctions

• Definition and meaning of disjunction

• Negation

• Negation of compound statements

• Negation of conjunction

• Negation of disjunction

• Negation of a negation

• The conditional statement

• The contrapositive of a conditional statement

• The converse of a conditional statement

• The biconditional statement

• Quantifiers

• Validity of statements

• Validity of the statement with “If and Only If”

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