Which of the following is/are not true ? (a) The set of negative integers is…

Reasoning: A set is countable if its elements can be put into a one-to-one correspondence with the natural numbers (i.e., listed in a sequence).

• The set of negative integers is countable. Example bijection: map each positive integer n to -n to list all negative integers.

• The set of integers that are multiples of 7 is countable. Example bijection: map each integer n to 7n to list all multiples of 7.

• The set of even integers is countable. Example bijection: map each integer n to 2n to list all even integers.

• The set of real numbers between 0 and 1/2 is uncountable. By Cantor's diagonal argument, any attempted listing of those real numbers misses infinitely many of them, so they cannot be placed in one-to-one correspondence with the natural numbers.

Conclusion: The only false claim is that the real numbers between 0 and 1/2 are countable. The other three sets are countably infinite, so the correct answer is the choice that identifies the real interval as not countable.