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Question

Let A and B be sets. If AX=BX=Φ and AX=BX for some set X. Show that A=B.

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Solution

Here AX=BX for some X

A(AX)=A(BX)

A=(AB)(AX) [ A(AX)=A]

A=(AB)Φ [ AX=Φ]

A=AB

AB ...(i)

Also AX=BX

B(AX)=B(BX)

(BA)(BX)=B [ B(BX)=B]

(BA)Φ=B [ BX=Φ]

BA=B

BA ...(ii)

From (i) and (ii), we have A=B


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