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.

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Solution

The contradiction of a statement is called the negation of a statement. The phrases like here exists” and “for every” are the quantifiers.

(i)

The given statement is: There exists a number which is equal to its square.

The quantifier in the given statement is “there exists”.

The negation of the given statement is: There does not exist a number which is equal to its square.

(ii)

The given statement is: For every real number x, x is less than x+1 .

The quantifier in the given statement is “for every”.

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

(iii)

The given statement is: There exists a capital for every state in India.

The quantifier in the given statement is “there exists”.

The negation of the given statement is: There exists a state in India which does not have a capital.

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