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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

Let A and B be two sets such that AX=BX=ϕ and AX=BX for some set X.

To show: A=B
It can be seen that
A=A(AX)
=A(BX) (AX=BX)
=(AB)(AX) (Distributive law)
=(AB)ϕ (AX=ϕ)
=AB ......(1)

Now, B=B(BX)
=B(AX) (AX=BX)
=(BA)(BX) (Distributive law)
=(BA)ϕ (BX=ϕ)
=BA
=AB ......(2)
Hence, from (1) and (2), we get
A=B.

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