# NCERT Class 11-Math՚S: Chapter – 14 Mathematical Reasoning Part 4 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**14.1. 15 The biconditional statement** If two statements and are connected by the connective ‘if and only if’ then the resulting compound statement “ if and only if ” is called a *biconditional* of *p* and *q* and is written in symbolic form as .

**Example 17**: Form the biconditional of the following statements:

One is less than seven

Two is less than eight

**Solution**: The biconditional of *p* and *q* is given by “One is less than seven, if and only if two is less than eight” .

**Example 18**: Translate the following biconditional into symbolic form: “ABC is an equilateral triangle if and only if it is equiangular” .

**Solution**: Let ABC is an equilateral triangle

and ABC is an equiangular triangle.

Then, the given statement in symbolic form is given by .

**14.1. 16 Quantifiers**: Quantifiers are the phrases like ‘These exist’ and “for every” . We come across many mathematical statement containing these phrases. For example – Consider the following statements

For every prime number , is an irrational number.

There exists a triangle whose all sides are equal.

**14.1. 17 Validity of statements**: Validity of a statement means checking when the statement is true and when it is not true. This depends upon which of the connectives, quantifiers and implication is being used in the statement.

(i) Validity of statement with ‘AND’

To show statement is true, show statement ‘’ is true and statement ‘’ is true.

(ii) Validity of statement with ‘OR’

To show statement is true, show either statement ‘’ is true or statement ‘’ is true.

(iii) Validity of statement with “If-then”

To show statement *r*: “If *p* then *q* is true” , we can adopt the following methods:

(a) Direct method: Assume *p* is true and show *q* is true, i.e.. , .

(b) Contrapositive method: Assume is true and show is true, i.e.. , .

(c) Contradiction method: Assume that *p* is true and *q* is false and obtain a contradiction from assumption.

(d) By giving a counter example: To prove the given statement *r* is false we give a counter example. Consider the following statement.

“: All prime numbers are odd” . Now the statement ‘’ is false as is a prime number and it is an even number.

**14.1. 18 Validity of the statement with “If and only If”** To show the statement if and only if *q* is true, we proceed as follows:

**Step 1**: Show if *p* is true then *q* is true.

**Step 2**: Show if *q* is true then *p* is true.

## 14.2 Solved Examples

## Short Answer Type

**Question1**: Which of the following statements are compound statements

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

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

(iii) “Ram and Rahim are friends”

**Answer**:

(i) The given statement can be broken into two simple statements “2 is an even number” and “2 is a prime number” and connected by the connective ‘and’

(ii) The given statement can be broken into two simple statements “9 is not an even number” and “9 is not a prime number” and connected by the connective ‘and’

(iii) The given statement can-not be broken into two simple statements and hence it is not a compound statement.

**Question 2**: Identify the component statements and the connective in the following compound statements.

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

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

**Answer**:

(a) The component statements are given by

It is raining

The sun is shining

The connective is “or”

(b) The component statements are given by

is a positive number

is a negative number

The connective is ‘or’

**Question 3**: Translate the following statements in symbolic form

(i) 2 and 3 are prime numbers

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

**Solution**

(i) The given statement can be rewritten as “ is a prime number and 3 is a prime number” .

Let is a prime number

is a prime number

Then the given statement in symbolic form is .

(ii) The given statement can be rewritten as

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

Let Tigers are found in Gir forest

Tigers are found in Rajaji national park.

Then the given statement in symbolic form is .