## EXERCISE 14.6

#### Page No 345:

#### Question 1:

Write the negation of the following statements:

(i) *p*: For every positive real number *x*, the number *x* – 1 is also positive.

(ii) *q*: All cats scratch.

(iii) *r*: For every real number *x*, either *x* > 1 or *x* < 1.

(iv) *s*: There exists a number *x* such that 0 < *x* < 1.

#### Answer:

(i) The negation of statement *p* is as follows.

There exists a positive real number *x*, such that *x* – 1 is not positive.

(ii) The negation of statement *q* is as follows.

There exists a cat that does not scratch.

(iii) The negation of statement *r* is as follows.

There exists a real number *x*, such that neither *x* > 1 nor *x* < 1.

(iv) The negation of statement *s* is as follows.

There does not exist a number *x*, such that 0 < *x* < 1.

#### Question 2:

State the converse and contrapositive of each of the following statements:

(i) *p*: A positive integer is prime only if it has no divisors other than 1 and itself.

(ii) *q*: I go to a beach whenever it is a sunny day.

(iii) *r*: If it is hot outside, then you feel thirsty.

#### Answer:

(i) Statement *p* can be written as follows.

If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is as follows.

If a positive integer has no divisors other than 1 and itself, then it is prime.

The contrapositive of the statement is as follows.

If positive integer has divisors other than 1 and itself, then it is not prime.

(ii) The given statement can be written as follows.

If it is a sunny day, then I go to a beach.

The converse of the statement is as follows.

If I go to a beach, then it is a sunny day.

The contrapositive of the statement is as follows.

If I do not go to a beach, then it is not a sunny day.

(iii) The converse of statement *r* is as follows.

If you feel thirsty, then it is hot outside.

The contrapositive of statement *r* is as follows.

If you do not feel thirsty, then it is not hot outside.

#### Question 3:

Write each of the statements in the form “if *p*, then *q*”.

(i) *p*: It is necessary to have a password to log on to the server.

(ii) *q*: There is traffic jam whenever it rains.

(iii) *r*: You can access the website only if you pay a subscription fee.

#### Answer:

(i) Statement *p* can be written as follows.

If you log on to the server, then you have a password.

(ii) Statement *q* can be written as follows.

If it rains, then there is a traffic jam.

(iii) Statement *r* can be written as follows.

If you can access the website, then you pay a subscription fee.

#### Question 4:

Re write each of the following statements in the form “*p* if and only if *q*”.

(i) *p*: If you watch television, then your mind is free and if your mind is free, then you watch television.

(ii) *q*: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.

(iii) *r*: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

#### Answer:

(i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if and only if it is a rectangle.

#### Question 5:

Given below are two statements

*p*:* 25 is a multiple of 5.*

*q: 25 is a multiple of 8.*

Write the compound statements connecting these two statements with “And” and “Or”. In both cases check the validity of the compound statement.

#### Answer:

The compound statement with ‘And’ is “25 is a multiple of 5 and 8”.

This is a false statement, since 25 is not a multiple of 8.

The compound statement with ‘Or’ is “25 is a multiple of 5 or 8”.

This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5.

#### Question 6:

Check the validity of the statements given below by the method given against it.

(i) *p*: The sum of an irrational number and a rational number is irrational (by contradiction method).

(ii) *q*: If *n* is a real number with *n* > 3, then *n*^{2} > 9 (by contradiction method).

#### Answer:

(i) The given statement is as follows. *p*: the sum of an irrational number and a rational number is irrational.

Let us assume that the given statement, *p*, is false. That is, we assume that the sum of an irrational number and a rational number is rational.

Therefore, , whereis irrational and *b*, *c*, *d*, *e* are integers.

⇒ de – bc = aBut here, is a rational number andis an irrational number.

This is a contradiction. Therefore, our assumption is wrong.

Therefore, the sum of an irrational number and a rational number is rational.

Thus, the given statement is true.

(ii) The given statement, *q*, is as follows.

If *n* is a real number with *n* > 3, then *n*^{2} > 9.

Let us assume that *n* is a real number with *n* > 3, but *n*^{2} > 9 is not true.

That is, *n*^{2} < 9

Then, *n* > 3 and *n* is a real number.

Squaring both the sides, we obtain

*n*^{2} > (3)^{2}

⇒ *n*^{2} > 9, which is a contradiction, since we have assumed that *n*^{2} < 9.

Thus, the given statement is true. That is, if *n* is a real number with *n* > 3, then *n*^{2} > 9.

#### Question 7:

Write the following statement in five different ways, conveying the same meaning.

*p: If triangle is equiangular, then it is an obtuse angled triangle.*

#### Answer:

The given statement can be written in five different ways as follows.

(i) A triangle is equiangular implies that it is an obtuse-angled triangle.

(ii) A triangle is equiangular only if it is an obtuse-angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse-angled triangle.

(iv) For a triangle to be an obtuse-angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is not an obtuse-angled triangle, then the triangle is not equiangular.