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Question

Let A and B be sets, If AX=BX=ϕ and AX=BX for some set X, prove that A=B.

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Solution

AX=BX for some set X
A(AX)=A(BX)
(AA)(AX)=A(BX)
A=(AB)(AX)
A=(AB)ϕ [AX=ϕ (given)]
A=AB
AB .......(i)
Again, AX=BX
B(AX)=B(BX)
(BA)(BX)=(BB)(BX)
(BA)(BX)=B
(BA)ϕ=B [BX=ϕ (given)]
BA=B
BA .........(ii)
From (i) and (ii) A=B.

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