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Question

For any two sets A and B, prove that

(i) BAB

(ii) ABA

(iii) ABAB=A

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Solution

(i) BAB

Let xϵA

x ϵ Aorx ϵ Bx ϵ AB

Hence , BAB

(ii) ABA

Let x ϵAB

x ϵ A and x ϵ B x ϵA

Hence, ABA

(iii) ABAB=A

If AB

Let x ϵ AB

x ϵ Aor x ϵ B x ϵB [AB]

ABB ....(i)

But BAB ....(ii)

From Eqs. (i) and (ii)

AB=B

If AB=B

Let y ϵ A

y ϵ ABy ϵ B [AB=B]

AB

Hence, ABAB=B


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