NCERT Solutions for Class 11 Maths Miscellaneous Exercise Chapter 1 - Sets

Miscellaneous Exercise

1. Decide among the following sets, which sets are the subsets of one and another:

$A=\left\{ x:x\in R\text{ satisfy }{{\text{x}}^{2}}-8x+12=0 \right\}$

$B=\left\{ 2,4,6 \right\}$

$C=\left\{ 2,4,6,8,... \right\},D=\left\{ 6 \right\}$

Ans: Given that,

$A=\left\{ x:x\in R\text{ satisfy }{{\text{x}}^{2}}-8x+12=0 \right\},B=\left\{ 2,4,6 \right\}$

$C=\left\{ 2,4,6,8,... \right\},D=\left\{ 6 \right\}$

$A=\left\{ x:x\in R\text{ satisfy }{{\text{x}}^{2}}-8x+12=0 \right\}$

${{x}^{2}}-8x+12=0$

$\left( x-2 \right)\left( x-6 \right)=0$

$x=2,6$

$A=\left\{ 2,6 \right\},B=\left\{ 2,4,6 \right\},$  $C=\left\{ 2,4,6,8,... \right\},D=\left\{ 6 \right\}$

A set A is said to be a subset of B if every element of A is also an element of B

$A\subset B$ if $a\in A,a\in B$

We can observe that,

$D\subset A\subset B\subset C$

$\therefore D\subset A,B,C\text{ and A}\subset B,C\text{ and B}\subset C$

2. In each of the following statements, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.

i. If $x\in A\text{ and A}\in \text{B, then x}\in \text{B}$

Ans: $x\in A\text{ and A}\in \text{B, then x}\in \text{B}$

To determine whether the statement is true

The given statement is false

For example,

Let $A=\left\{ 1,2 \right\},B=\left\{ 1,\left\{ 1,2 \right\},\left\{ 3 \right\} \right\}$

Now $2\in \left\{ 1,2 \right\},\left\{ 1,2 \right\}\in \left\{ 1,\left\{ 1,2 \right\},\left\{ 3 \right\} \right\}$

But $2\notin \left\{ 1,\left\{ 1,2 \right\},\left\{ 3 \right\} \right\}$

Hence the given statement is false.

ii. If $A\subset B\text{ and }B\in C,thenA\in C$

Ans: $A\subset B\text{ and }B\in C,thenA\in C$

To determine whether the statement is true

The statement is false

For example,

Let $A=\left\{ 2 \right\},B=\left\{ 0,2 \right\},C=\left\{ 1,\left\{ 0,2 \right\},3 \right\}$

As $A\subset B,B\in C$ but $A\notin C$

iii. If $A\subset B\text{ and }B\subset C,thenA\subset C$

Ans:

$A\subset B\text{ and }B\subset C,thenA\subset C$

To determine whether the statement is true

The given statement is true.

Let $A\subset B,B\subset C$

Let $x\in A$ 

$x\in B$so $A\subset B$

$x\in C$so $B\subset C$

iv. If $A\not\subset B\text{ and }B\not\subset C,thenA\not\subset C$

Ans: $A\not\subset B\text{ and }B\not\subset C,thenA\not\subset C$

To determine whether the statement is true

The given statement is false

Let $A=\left\{ 1,2 \right\},B=\left\{ 0,6,8 \right\},C=\left\{ 0,1,2,6,9 \right\}$

Now by the statement,

$A\not\subset B\text{ and }B\not\subset C$

But $A\subset C$

v. If $x\in A\text{ and A}\not\subset \text{B, then x}\in \text{B}$

Ans: $x\in A\text{ and A}\not\subset \text{B, then x}\in \text{B}$

To determine whether the statement is true

The given statement is false

Let $A=\left\{ 3,5,7 \right\},B=\left\{ 3,4,6 \right\}$

Now, $5\in A$ and $A\not\subset B$

But $5\notin B$

vi. If $\text{A}\subset \text{B and x}\notin \text{B, then x}\notin \text{A}$

Ans: $\text{A}\subset \text{B and x}\notin \text{B, then x}\notin \text{A}$

To determine whether the statement is true

The given statement is true

Let $A\subset B$ and $x\notin B$

To show that $x\notin A$

Suppose $x\in A$

Then $x\in B$ which is a contradiction

$\therefore x\notin A$

3. Let $A,B$ and $C$ be the set such that $A\cup B=A\cup C$ and $A\cap B=A\cap C$. Show that $B=C$

Ans: To show that $B=C$

Let $x\in B$

$x\in A\cup B$ since $\left[ B\subset A\cup B \right]$

$x\in A\cup C$ since $\left[ A\cup B=A\cup C \right]$

$x\in A\text{ or }\in \text{C}$

Case I:

Also $x\in B$

$x\in A\cap B$ since $\left[ B\subset A\cap B \right]$

$x\in A\cap C$ since $\left[ A\cap B=A\cap C \right]$

$x\in A\text{ or x}\in \text{C}$

$\therefore x\in C$

$\therefore B\subset C$

Similarly, we can show that $C\subset B$

$\therefore $It has been proved that $B=C$

4. Show that the following four conditions are equivalent:

i. $A\subset B$

ii. $A-B=\varnothing $

iii. $A\cup B=B$

iv. $A\cap B=A$

Ans: To show that the above four conditions are equivalent

First showing that,

(i)⬄(ii),

Let $A\subset B$

To show,

$A-B=\varnothing $

If possible, suppose,

$A-B\ne \varnothing $

This means that there exist $x\in A,x\notin B$, which is impossible as $A\subset B$

$\therefore A-B=\varnothing $

$A\subset B$=> $A-B=\varnothing $

Let $A-B=\varnothing $

To show that,

$A\subset B$

Let $x\in B$ since if $x\notin B,$ then $A-B\ne \varnothing $

$\therefore $  $A-B=\varnothing $ => $A\subset B$

$\therefore $(i)⬄ (iii)

Let $A\subset B$

To show $A\cup B=B$

Clearly$B\subset A\cup B$

Let $x\in A\cup B$

$x\in A$ or $x\in B$

Case I:

$x\in A$

So that $x\in B$

$\therefore A\cup B\subset B$

Case II:

$x\in B$

Then $A\cup B=B$

Conversely let $A\cup B=B$

Let $x\in A$

$x\in A\cup B$ since $\left[ A\subset A\cup B \right]$

$x\in B$ since $A\cup B=B$

$\therefore A\subset B$

Hence (i) ⬄ (iii)

Now we have to show that (i) ⬄ (iv)

Let $A\subset B$

$A\cap B\subset A$

Let $x\in A$

We have to show that $x\in A\cap B$

As $A\subset B$, $x\in B$

$x\in A\cap B$

$A\subset A\cap B$

Hence $A=A\cap B$

Let $x\in A$

$x\in A\cap B$

$x\in A$ and $x\in B$

So that $x\in B$

$\therefore A\subset B$

Hence (i) ⬄ (iv)

5. Show that if$A\subset B$, then $C-B\subset C-A$

Ans: Given that,

$A\subset B$

To show that,

$C-B\subset C-A$

Let $x\in C-B$

$x\in C$ and $x\in B$

$x\notin A\left[ A\subset B \right]$ and $x\in C$

$x\in C-A$

$\therefore C-B\subset C-A$

Hence it has been showed that $C-B\subset C-A$

6. Show that for any sets $A$ and$B$,

$A=\left( A\cap B \right)\cup \left( A-B \right)$ and $A\cup \left( B-A \right)=A\cup B$

Ans: To show that,

$A=\left( A\cap B \right)\cup \left( A-B \right)$

Let $x\in A$

We have to show that $A=\left( A\cap B \right)\cup \left( A-B \right)$

Case I:

$x\in A\cap B$

Then $x\in A\cap B\subset \left( A\cup B \right)\cup \left( A-B \right)$

Case II:

$x\notin A\cap B$

$x\notin A$ or $x\notin B$

$x\notin B\left[ x\notin A \right]$

$x\notin A-B\subset \left( A\cup B \right)\cup \left( A-B \right)$

$\therefore A\subset \left( A\cap B \right)\cup \left( A-B \right)$                                          ……(1)

It is clear that,

$A\cap B\subset A$ and $A-B\subset A$

$\therefore \left( A\cap B \right)\cup \left( A-B \right)\subset A$                                          ……(2)

From (1) and (2) we obtain that,

$A=\left( A\cap B \right)\cup \left( A-B \right)$

To prove that,

$A\cup \left( B-A \right)\subset A\cup B$

Let $x\in A\cup \left( B-A \right)$

$x\in A$ or ($x\in B$ and $x\notin A$)

($x\in A$ or $x\in B$) and ($x\in A$ or $x\notin A$)

$x\in A\cup B$

$\therefore A\cup \left( B-A \right)\subset A\cup B$                                             ……(3)

Next we show that $A\cup \left( B-A \right)\subset A\cup B$

Let $y\in A\cup B$

$y\in A$ or $y\in B$

($y\in A$ or $y\in B$) and ($y\in A$or $y\notin A$)

$y\in A\cup \left( B-A \right)$

$\therefore A\cup \left( B-A \right)\subset A\cup B$

Hence from (3) and (4) we obtain that$A\cup \left( B-A \right)=A\cup B$

Hence proved the statement

7. Using properties of sets, show that

i. $A\cup \left( A\cap B \right)=A$

Ans: To show that, $A\cup \left( A\cap B \right)=A$ by using the property offsets

We know that,

$A\subset A$

$A\cap B\subset A$

$\therefore A\cup \left( A\cap B \right)\subset A$                             ……(1)

And $A\subset A\cup \left( A\cap B \right)$                         ……(2)

From (1) and (2) we get that

$A\cup \left( A\cap B \right)=A$

Hence it has been showed that $A\cup \left( A\cap B \right)=A$

ii. $A\cap \left( A\cup B \right)=A$

Ans: To show that $A\cap \left( A\cup B \right)=A$ by using the property of sets

$A\cap \left( A\cup B \right)=\left( A\cap A \right)\cup \left( A\cap B \right)$

$=A\cup \left( A\cap B \right)$

$=A$(from (1))

Hence it has been shown that $A\cap \left( A\cup B \right)=A$

8. Show that $A\cap B=A\cap C$ need not imply $B=C$

Ans: Given that,

$A\cap B=A\cap C$

To show that the above does not imply $B=C$

Let $A=\left\{ 0,1 \right\},B=\left\{ 0,2,3 \right\},C=\left\{ 0,4,5 \right\}$

Now, 

$A\cap B=\left\{ 0 \right\}=A\cap C$

But $B\ne C$ since $2\in B$ and not to $C$

$\therefore $It has been shown that $A\cap B=A\cap C$ does not necessarily imply $B=C$

9. Let $A$ and $B$ be sets. If $A\cap X=B\cap X=\varnothing $ and $A\cup X=B\cup X$ for some text $X$, show that $A=B$

(Hints: $A=A\cap \left( A\cup X \right),B=B\cap \left( B\cup X \right)$ and use distributive law)

Ans: Given that,

$A\cap X=B\cap X=\varnothing $

$A\cup X=B\cup X$

To show that,

$A=B$

We know that,

$A=A\cap \left( A\cup X \right)$

$=A\cap \left( B\cup X \right)$ since $A\cup X=B\cup X$

$=\left( A\cap B \right)\cup \left( A\cap X \right)$ By distributive law

$=\left( A\cap B \right)\cup \varnothing $ since $A\cap X=\varnothing $

$=A\cap B$                                              ……(1)

Now consider$B=B\cap \left( B\cup X \right)$

$=B\cap \left( A\cup X \right)$ since $B\cup X=A\cup X$

$=\left( B\cap A \right)\cup \left( B\cap X \right)$ By distributive law

$=\left( B\cap A \right)\cup \varnothing $ since $B\cap X=\varnothing $

$=B\cap A$

$=A\cap B$                                               ……(2)

It is possible only when $A=B$

So from (1) and (2), we get,

$A=B$

10. Find sets $A,B$ and $C$ such that $A\cap B,B\cap C$ and $A\cap C$ are non-empty subsets and $A\cap B\cap C=\varnothing $

Ans: Given that,

$A\cap B,B\cap C,A\cap C$ are non-empty subsets and

$A\cap B\cap C=\varnothing $

Let $A=\left\{ 0,2 \right\},B=\left\{ 1,2 \right\},C=\left\{ 2,0 \right\}$

Now, 

$A\cap B=\left\{ 1 \right\}$

$B\cap C=\left\{ 1,2 \right\}$

$A\cap C=\left\{ 0 \right\}$

And $A\cap B\cap C=\varnothing $

$\therefore $It has been showed that $A\cap B,B\cap C$ and $A\cap C$ are non-empty subsets but however $A\cap B\cap C=\varnothing $

Conclusion

NCERT Miscellaneous Exercise Class 11 Chapter 1 Maths is crucial for understanding various concepts thoroughly. It covers diverse problems that require the application of multiple formulas and techniques. It's important to focus on understanding the underlying principles behind each question rather than just memorizing solutions. Remember to understand the theory behind each concept, practice regularly, and refer to solved examples to master this exercise effectively.

Class 11 Maths Chapter 1: Exercises Breakdown

CBSE Class 11 Maths Chapter 1 Other Study Materials

Chapter-Specific NCERT Solutions for Class 11 Maths

Given below are the chapter-wise NCERT Solutions for Class 11 Maths. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

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