**Question 1.**

**For each of the following compound statements first, identify the connecting words and then break them into component statements.**

**(i)** All rational numbers are real and all real numbers are not complex.

**(ii)** Square of an integer is positive or negative.

**(iii)** The sand heats up quickly in the Sun and does not cool down fast at night.

**(iv)** x = 2 and x = 3 are the roots of the equation 3x^{2} – x – 10 = 0.

**Solution:**

**(i)** The compound statement has the connecting word ‘and’. Component statements are

p: All rational numbers are real.

q: All real numbers are not complex.

**(ii)** The compound statement has the connecting word ‘or’. Component statements are:

p: Square of an integer is positive.

q: Square of an integer is negative.

**(iii)** The compound statement has the connecting word ‘and’. Component statements are:

p: The sand heats up quickly in the sun.

q: The sand does not cool down fast at night.

**(iv)** The compound statement has the connecting word ‘and’. Component statements are:

p: x- 2 is a root of the equation 3x^{2} – x – 10 = 0.

q: x = 3 is a root of the equation 3x^{2} – x – 10 = 0.

**Question 2.**

**Identify the quantifier in the following statements and write the negation of the statements.**

**(i)** There exists a number that is equal to its square.

**(ii) **For every real number x, x is less than x + 1.

**(iii)** There exists a capital for every state in India.

**Solution:**

**(i)** Here the quantifier is ‘there exists’.

** The negation of a statement is: **There does not exist a number that is equal to its square.

**(ii)** Here the quantifier is ‘for every’

** The negation of a statement is:** For at least one real number x, x is not less than x + 1.

**(iii)** Here the quantifier is ‘there exists’

** The negation of a statement is:** There exists a state in India that does not have capital.

**Question 3.**

**Check whether the following pair of statements is the negation of each other. Give reasons for your answer.**

**(i)** x + y = y + x is true for every real numbers x and y.

**(ii)** There exists real numbers x and y for which x + y = y + x.

**Solution:**

Let p: x + y = y + x is true for every real numbers x and y.

q: There exists real numbers x and y for which

x+y=y + x.

Now, ~p: There exist real numbers x and y for which x + y ≠ y + x.

Thus, ~p ≠ q.

**Question 4.**

**State whether the “Or” used in the following statements is “exclusive” or “inclusive”. Give reasons for your answer.**

**(i)** Sunrises or Moon sets.

**(ii)** To apply for a driving license, you should have a ration card or a passport.

**(iii)** All integers are positive or negative.

**Solution:**

**(i)** This statement makes use of exclusive “or”. Since when sunrises, the moon does not set during the daytime.

**(ii)** This statement makes use of inclusive ‘or’. Since you can apply for a driving license even if you have a ration card as well as a passport.

**(iii)** This statement makes use of exclusive ‘or’. Since an integer is either positive or negative, it cannot be both.