### Mathematical Reasoning Exercise 03

## Question:

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

## Answer:

Answer: No, the statements are not negations of each other.

The first statement is an assertion that the equation x+y=y+x is true for every real numbers x and y.

The second statement is a statement that there exists real number x and y for which the equation x+y=y+x is true.

These two statements are not negations of each other because they are not contradicting each other.

## Question:

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 licence you should have a ration card or a passport (iii) All integers are positive or negative

## Answer:

(i) Exclusive. The statement implies that only one of the two events (Sun rising or Moon setting) can occur.

(ii) Inclusive. The statement implies that either a ration card or a passport can be used to apply for a driving licence.

(iii) Exclusive. The statement implies that only one of the two conditions (all integers being positive or negative) can be true.

## Question:

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

## Answer:

(i) Quantifier: There exists Negation: There does not exist a number which is equal to its square.

(ii) Quantifier: For every Negation: There exists a real number x such that x is not less than x+1.

(iii) Quantifier: There exists Negation: There does not exist a capital for every state in India.

## Question:

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 3x2−x−10=0

## Answer:

(i) Connecting words: and Component statements: All rational numbers are real; All real numbers are not complex.

(ii) Connecting words: or Component statements: Square of an integer is positive; Square of an integer is negative.

(iii) Connecting words: and Component statements: The sand heats up quickly in the Sun; The sand does not cool down fast at night.

(iv) Connecting words: and Component statements: x=2 is a root of the equation 3x2−x−10=0; x=3 is a root of the equation 3x2−x−10=0.