**Question 1: Write the negation of the following statements:**

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

**Solution:**

~p: There exists atleast a positive real number x, such that x – 1 is not positive.

**(ii) q: All cats scratch. **

**Solution:**

~q: There exists cats that do not scratch.

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

**Solution:**

~r: There exists a real number x, such that neither x > 1 nor x < 1.

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

**Solution:**

~s: 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.**

**Solution:**

Statement p can be understood 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 divisor other than 1 and itself, then it is prime.

The contrapositive of the statement is as follows.

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

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

**Solution:**

The given statement can be understood 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) r: If it is hot outside, then you feel thirsty. **

**Solution:**

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. **

**Solution:**

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

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

**Solution:**

q: If it rains, then there is a traffic jam.

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

**Solution:**

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

**Question 4: Rewrite 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.**

**Solution:**

p: You watch television if and only if your mind is free.

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

**Solution:**

q: You get an A grade if and only if 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.**

**Solution:**

r: 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. **

**Solution:**

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

This statement is not valid, because 25 is not a multiple of 8.

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

This statement is valid, because although 25 is not a multiple of 8, 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). **

**Solution:**

The given statement is,

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, √a + b/c = d/e is irrational where a, b, c, d and e are integers.

d/e – b/c is a rational number and √a is an irrational number.

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

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

Hence, the given statement is valid.

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

^{2}> 9 (by contradiction method).

**Solution:**

The given statement is,

q: If n is a real number with n > 3, then n

^{2}> 9.Let’s assume that n is a real number with n > 3, but n

^{2}> 9 is false,i.e., n^{2 }< 9.Then, n > 3 where n is a real number.

Squaring both the sides,

n

^{2}> (3)^{2}⇒ n

^{2}> 9, which is a contradiction to our assumption that is n^{2}< 9.Therefore, the given statement is valid.

**Question 7: Write the following statement in five different ways, conveying the same meaning.**

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

**Solution:**

The given statement can be written in the following five different ways:

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

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

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

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

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