Write the negation of the following statements:(i) p: For every positi

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: For every real number x, either x > 1 or x < 1.(iii) r: There exists a number x such that 0 < x < 1.
Answer» (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.

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»

(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.

~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.

~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.

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