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) 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 also positive.

(ii) 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) The negation of the statement :

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

is

~r: For every real number x , either $x\le orx\ge 1.$