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Question

On the power set P of a non-empty set A, we define an operation ∆ by XY=XY XY
Then which are of the following statements is true about ∆
(a) commutative and associative without an identity
(b) commutative but not associative with an identity
(c) associative but not commutative without an identity
(d) associative and commutative with an identity

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Solution

(d) associative and commutative with an identity

Commutativity:XY=XYXY =YXYX =YXThus, XY=YXHence, is commutative on A.

Let ϕ be the identity element for on P.
Aϕ=AϕAϕ =ϕA =Aand, ϕA=ϕAϕA =Aϕ =A

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