**For each of the following compound statements first identify the connecting words and then break it 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 connecting word in statement1 is ‘and’.

The component statements are as follows.

a: All rational numbers are real.

b: All real numbers are not complex.

(ii) The connecting word in statement1 is ‘or’.

The component statements are as follows.

a: Square of an integer is positive.

b: Square of an integer is negative.

(iii) The connecting word in statement1 is ‘and’.

The component statements are as follows.

a: The sand heats up quickly in the sun.

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

(iv) The connecting word in statement1 is ‘and’.

The component statements are as follows.

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

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

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

**(i) There exists a number which 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) The quantifier is “There exists”.

The negation of this statement is: “There does not exist a number which is equal to its square.”

(ii)The quantifier is “For every”.

The negation of this statement is: “There exist a real number x such that x is not less than x + 1.”

(iii) The quantifier is “There exists”.

The negation of this statement is: “There exists a state in India which does not have a capital.”

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

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

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

Solution:

The negation of statement (i) is as follows.

There exists real number x and y for which x + y ≠ y + x. This is not the same as statement

(ii). Thus, the given statements are not the negation of each other.

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

**(i) Sun rises 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) Here, “or” is exclusive because it is not possible for the Sun to rise and the moon to set together.

(ii)Here, “or” is inclusive since a person can have both a ration card and a passport to apply for a driving license.

(iii) Here, “or” is exclusive because all integers cannot be both positive and negative.