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Let X be a set and define the binary operation ∆ on the subsets of X by A∆B := (A∩Bc)∪(B∩Ac).Prove that the set P(X) of all subsets of X is a group under the operation ∆.

Question
  1. Let X be a set and define the binary operation ∆ on the subsets of X by A∆B := (A∩Bc)∪(B∩Ac).

    Prove that the set P(X) of all subsets of X is a group under the operation ∆.

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Step 1

Let X be a set and define the binary operation ∆ on the subsets of X by

AAB-(AnB(Bn4)
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AAB-(AnB(Bn4)

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Step 2

Prove that the set P(X) of all subsets of X is a group under the operation ∆.

Proof:

Consider the binary structure (P(x),A).
Verify whether the Identity exists or not
Let EP(X) then for A e P(X); the operationbecomes
A -(øn4)(n)
=
=ØUA
= A
Thus, the identity is Ø that exists in P(X)
help_outline

Image Transcriptionclose

Consider the binary structure (P(x),A). Verify whether the Identity exists or not Let EP(X) then for A e P(X); the operationbecomes A -(øn4)(n) = =ØUA = A Thus, the identity is Ø that exists in P(X)

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Step 3

Now, verify whether the Inv...

Let A EP(X) for AEP(X); the operation becomes
A'A4 (A)4n(4°)
(An)U(An4)
P(x)
Thus, the inverse is A that exists in P(x)
help_outline

Image Transcriptionclose

Let A EP(X) for AEP(X); the operation becomes A'A4 (A)4n(4°) (An)U(An4) P(x) Thus, the inverse is A that exists in P(x)

fullscreen

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