Question

B1: ({0, 1, 2….(n-1)}, xm) where xn stands for “multiplication-modulo-n” and B2: ({0, 1, 2….n}, xn) where xn stands for “multiplication-modulo-m” are the two statements. Both B1 and B2 are considered to be __________

(a) groups

(b) semigroups

(c) subgroups

(d) associative subgroup

This question was addressed to me in a job interview.

The question is from Groups in section Groups of Discrete Mathematics

Answer 1

The correct choice is (b) semigroups

Easy explanation: Here, B1 is the group and identity element is 0, means for all a∈B1, a+n.0=a. As a<n. But in B2 identity element does not exist. Here, 0 can not be the identity element. For example, for one of the member n of the set we have n+n.0=0, It will be n. So, B2 is not a group. Both B1 and B2 are semigroups as they satisfy closure and associativity property.