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: For every real number x, either x > 1 or x < 1.

(iii) r: There exists a number x such that 0 < x < 1.

Answer 1

(i) p : For every positive real number x, the number (x – 1) is also positive.

The negation of the statement:

p: For every positive real number x, the number (x – 1) is also positive.

is

~p: There exists a positive real number x, such that the number (x – 1) is not positive.

(ii) q: For every real number x, either x > 1 or x < 1.

The negation of the statement:

q: For every real number x, either x > 1 or x < 1.

is

~q: There exists a real number such that neither x>1 or x<1.

(iii) r: There exists a number x such that 0 < x < 1.

The negation of the statement:

r: There exists a number x such that 0 < x < 1.

is

~r: For every real number x, either x ≤ 0 or x ≥ 1.