Question

Write the negation of the following statements: (i) p : For every positive real number x , the number x – 1 is also positive. (ii) q : All cats scratch. (iii) r : For every real number x , either x > 1 or x < 1. (iv) s : There exists a number x such that 0 < x < 1.

Answer 1

The contradiction of a statement is called the negation of a statement.

(i) The given statement is: p: “For every positive real number x, the number x−1 is also positive.” The negation of the given statement is: There exists a positive real number x, such that x−1 is not positive.

(ii) The given statement is: q: “All cats scratch.” The negation of the given statement is: There exists a cat that does not scratch.

(iii) The given statement is: r: “For every real number x, either x>1 or x<1 .” The negation of the given statement is: There exists a real number x, such that neither x>1 nor x<1 .

(iv) The given statement is: s: “There exists a number x such that 0<x<1 .” The negation of the given statement is: There does not exists a number x such that 0<x<1 .