Fsc Part 1 Mathematics (Complete Solution)

Q8
Determine whether, (P(S),*),where * stands for intersection is a semi-group, a monoid or neither. If it is a monoid, specify its identity.

Solution:

let A,B∈ P(S) where A&B are subset of S.

As intersection of two subsets of S is subset of S.

Therefore A*(B*C)=A ∩ B ∈  P(S). Thus closure law holds in P(S).

For A,B,C∈ P(S)

A*(B*C)=A∩(B∩C)=(A∩B)∩C=(A*B)*C

Thus associative law holds and P(S)

And hence ( P(S),* ) is a semi-group.

For A∈P(S) where A is a subset of S we have S∈P(S) such that

A∩S=S∩A=A

Thus S is an identity element in P(S) . And hence ( P(S),* ) is a monoid.

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