Statements in Mathematical Reasoning

Your laptop is either ON or OFF. If it is not ON then surely it will be OFF. Isn't it? This is the basic Logic. The word ‘Logic’ you might have heard several times, but do you know what exactly is Logic? It is the study of basic patterns of reasoning. Logic does not include or refer to any particular meaning or context.

Mathematical reasoning is the concept in Mathematics that deals with finding the truth values of any Mathematical statement.  The principle of Mathematical reasoning is generally used to analyze the conceptual logical thinking capacity of an individual in competitive examinations and eligibility tests. Mathematical reasoning questions are extremely interesting and successfully stir up the rational thinking of the human brain. There are various kinds of statements in Mathematical reasoning and the operations that are performed on those statements. 

What is a Mathematical Reasoning Statement?

A Mathematical statement is a statement written in such a way that it can either be true or false but can never be both true and false simultaneously.  (Can you imagine your laptop being on both stages simultaneously?)

Before diving deep into the Mathematical Reasoning Statements, let us first understand which type of sentences could be considered as statements. For example, 

1. “New Delhi is the capital of India”. 

This sentence is true. Therefore this sentence is a statement. 

2. “There are 10 states in India”. 

This sentence is false. Therefore this sentence is a statement.

3. “Come here”.

This sentence is in the form of order, so this sentence has no logic. Therefore this sentence is not a statement. 

So, the sentences which give information either correct or incorrect at any instant, are the statements.

For a clear understanding of 'What is a Mathematical Reasoning Statement?’, let us consider two statements. 

Statement 1: “Sum of any two prime numbers is even”.

Statement 2: “Every square is a quadrilateral”.

In statement 1, it is given that when two prime numbers are added, the resultant number is an even number. Prime numbers are those numbers that are divisible by 1 and itself only. If we consider two prime numbers 2 and 3 as examples, the sum of 2 and 3 = 5 and the sum of 2 and 2 = 4. ‘5’ is an odd number whereas ‘4’ is an even number. So, we can conclude that the sum of any two prime numbers can be either even or odd. So statement 1 may be true or false at the same time. So it is not a statement according to Mathematics.

In statement 2, it is mentioned that every square is a quadrilateral. A quadrilateral is a polygon with 4 sides, 4 angles, and 4 vertices. A square also has 4 sides, 4 angles, and 4 vertices. So, it is always true forever that all squares are quadrilaterals. Because the statement is completely true at an instance, it is a Mathematical statement. 

Types of Maths Reasoning Statements

Maths reasoning statements are of 3 types namely 

1. Simple statements, 

2. Compound statements, and 

3. Conditional statements.

Simple Statements

Simple statements are those direct logical statements that do not contain any modifiers and cannot be further broken into two or simpler statements. 

Example: 1. “Earth is a planet”. 

2. “3 is an odd number.”

The above statements can be considered as a Math reasoning statement because they are always true. It is a simple statement because it cannot be broken down into further simple statements. 

Compound Statements

Compound statements are Mathematical statements that can be broken down into two or more simple statements. In other words, two or more simple statements are joined together with connectives to form a compound statement. 

Example: “Earth is a planet and has a natural satellite”. 

The above statement is a compound statement because it can be broken down into two simple statements “Earth is a planet” and “Earth has a natural satellite”. These two simple statements are joined together ‘and’ which is a connective. The two simple statements which together form a compound statement are called component statements.  

Connectives used in Compound Statements:

Connectives are the words that are used as a bridge to form a compound sentence by two more compound statements. There are three types of connectives that are used to combine simple statements to form a compound statement. Connectives are simple English words used to connect the statements. 

The Three Types of Connectives are:

1. Conjunction: A conjunctive statement is the compound statement that is obtained by connecting two simple statements with the connective ‘and’. The conjunction of two statements ‘p’ and ‘q’ is given as “p and q” or “p ∧ q”.

Example: The conjunction of “Time is money” and “Money is time” is given as “Time is money and money is time”.

2. Disjunction: A disjunctive statement is the compound statement that is obtained by joining two or more statements with a connecting word ‘or’. The disjunction of two statements ‘p’ and ‘q’ is given as “p or q” or “p ∨ q”.

Example: The disjunction of two statements “A natural number is an odd number” and “A natural number is an even number” is given as “A natural number is an odd number or an even number”. 

3. Negation: Negation of a statement is the statement representing denial of any statement. The word ‘not’ is used to negate a statement. Through negation does not join two statements, it gives the negative of a statement. Negation of any statement ‘p’ is given as “not p” and is symbolically represented as “~p”. 

Example: Consider a statement “7 is a prime number”. The negation of this statement is given as “7 is not a prime number”. 

Mathematical Reasoning Formulas used in Compound Statements:

If ‘p’ and ‘q’ are two Mathematical statements, then important Mathematical reasoning formulas are as follows.

• p ∧ q represents the conjunction of two statements indicating ‘p and q’.

• p ∨ q represents the disjunction of two statements indicating ‘p or q’.

• ~p represents the negation of a statement and indicates ‘not p’.

• The negation of a conjunction statement is given as the conjunction of the negation of two statements.

~ (p ∧ q) = ~ p ∧ ~ q  

• The negation of a disjunction statement is given as the disjunction of negation of both statements. 

~ (p ∨ q) = ~ p ∨ ~ q

Conditional Statements:

Conditional statements are compound statements in which the truth value of one statement depends on the truth value of the other statement. i.e., the second statement is true only if the first statement is true. A conditional statement with ‘p’ and ‘q’ as simple statements can be symbolically represented as “p → q” which represents “p if so q”. In this statement, p is called ‘antecedent’, and q is called the conclusion. 

Example: If there is no government holiday, then the man is available.

In the above example, “There is no government holiday” is the condition to be checked to find whether “The man is available”. So, the former is the antecedent and the latter is the conclusion. 

Inverse, Converse, and Contrapositive of Conditional statements:

1. Inverse: Inverse of a conditional statement indicates the statement representing the negation of antecedent and negation of conclusion expressed as a conditional statement.

Let an example of a conditional statement be  “If a number is less than zero, then it is negative”. 

The inverse of this statement is given as “If a number is not less than zero, then it is not negative”. 

2. Converse: Converse of a conditional statement is represented by swapping the antecedent and conclusion. 

For example, the converse of the statement “If it is night, then we have dinner” is written as “If we have dinner, then it is the night”. 

In the above example, it is clearly indicated that the term ‘dinner’ is used only for what we eat during the night. 

3. Contrapositive: Contrapositive of a conditional statement is another conditional statement that represents the negation of the conclusion of the first statement as the antecedent of the second and the negation of the antecedent of the first statement as the conclusion of the second.

Let us consider a conditional statement “If he knows swimming, then he can cross the river”. The contrapositive of this statement is written as “If he cannot cross the river, then he does not know swimming”. 

Mathematical Reasoning Formulas of Conditional Statements:

In any of the mathematical reasoning questions, a conditional statement is represented in the “if then” form as p → q where ‘p’ is the antecedent and ‘q’ is the conclusion.

The inverse of a conditional statement p → q is given as ~ p → ~ q.

The converse of a conditional statement p → q is given as q → p.

The contrapositive of a conditional statement p → q is given as ~ q → ~ p.

Mathematical Reasoning Questions:

Consider two statements represented as ‘a’ and ‘b’ where a = 2 + 3 = 6 and b = I am mad. Write a conditional statement indicating a → b. Write the inverse, converse, and contrapositive of a → b.

Solution:

The conditional statement a → b is written as

“If 2 + 3 = 6, then I am mad”.

The inverse of the above statement is: 

“If 2 + 3 ≠ 6, then I am not mad”.

The converse is written as:

“If I am mad, then 2 + 3 = 6”.

The contrapositive of the conditional statement is:

“If I am not mad, then 2 + 3 ≠ 6”.

Validity of the Statement

The validity of the statement is to check when the given statement is true or false. It depends upon the type of connectives, quantifiers, and implications used in the statement. 

1. Validity of statement with “AND”: If AND is used in a given statement, then both the sentences should be TRUE, to show the complete statement ‘TRUE’.

2. Validity of statement with “OR”: In this case, if any of the statements is TRUE, then the complete statement is considered valid or TRUE.

3. Validity of statement with “If-then”: To validate such statements TRUE, we either assume one sentence is true and prove another statement is also true or we assume one is FALSE and prove another statement also FALSE. 

4. Validity of the statement with “If and Only If”: For such statements to validate, we generally follow two-step procedure: 

Step 1: If the first sentence is TRUE, then the second sentence is true.

Step 2: After step 1, show if the second sentence is true, then the first sentence is also true. 

Do You Know What an Open Statement Is?

“X is a positive integer”. This statement is incomplete in itself and it depends on the value of x. It is not considered as a statement. However, it is called an open statement. 

A statement is either true or false, this truth and falsity are called truth value.

Points to Remember

A sentence can not be a statement if it is:

1. In exclamatory form.

2. In the form of a request or order.

3. In the form of a question.

4. Including timings such as today, yesterday or tomorrow, etc. 

5. Including variable places like there, here, etc.

6. Including pronouns like she, he, they, etc.

Fun Facts

When we apply negation to the negation of any statement, the result will be the statement itself. i.e. ~ ( ~p ) = p.

Phrases such as “for every” and “there exist” is used in Mathematical reasoning statements. They are called quantifiers.