Question

The set of odd and even positive integers closed under multiplication is ________

(a) a free semigroup of (M, ×)

(b) a subsemigroup of (M, ×)

(c) a semigroup of (M, ×)

(d) a subgroup of (M, ×)

I had been asked this question in an interview for job.

Origin of the question is Group Axioms topic in division Groups of Discrete Mathematics

Answer 1

Right choice is (b) a subsemigroup of (M, ×)

To explain: Let C and D be the set of even and odd positive integers. Then, (C, ×) and (D, ×) are subsemigroups of (M, ×) since A and B are closed under multiplication. On the other hand, (A, +) is a subsemigroup of (N, +) since A is closed under addition, but (B, +) is not a subsemigroup of (N, +) since B is not closed under addition.