Are the following two statements logically equivalent? Justify your answer.

Question

(a) A real number is less than 1 only if its reciprocal is greater than 1.
(b) Having a reciprocal greater than 1 is a sufficient condition for a real number to be less than 1.

Answer 1

These two statements are not logically equivalent.
Explanation 1: The first statement is equivalent to “If a real number is less than 1, then its reciprocal
is greater than 1” and the second statement is equivalent to “If the reciprocal of a real number is
greater than 1, then the number is less than 1.” Thus the second statement is the converse of the first,
and a conditional statement and its converse are not logically equivalent.

Explanation 2: The first statment is false. For example, ?2 is less than 1, but its reciprocal, ?1
2 is greater than 1. However, the second statement is true; if the reciprocal of a real number is greater than 1, then the number itself is positive and is between 0 and 1, and it is impossible for one of a pair of equivalent statements to be true while the other member of the pair is false.