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.

Question

Philoid · Accepted Answer

(i) The negation of the statement p is

It is false that for every positive real number x, the number x – 1 is also positive.

This can be rewritten as

There exists a positive real number x such that the number x – 1 is not positive.

(ii) The negation of the statement q is

It is false that all cats scratch.

This can be rewritten as

There is at least one cat that does not scratch.

(iii) The negation of the statement r is

It is false that for every real number x, either x > 1 or x < 1.

This can be rewritten as

There exists a real number x such that neither x > 1 nor x < 1.

(iv)The negation of the statement s is

It is false that there exists a number x such that 0 < x < 1.

This can be rewritten as

There does not exist a number x such that 0 < x < 1.

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.