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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»

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.



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