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Question

Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.

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Solution

The sets are given as,

AB=AC

AB=AC

Let,

xB

Here, one of the elements of set B is x.

So,

xAB xAC

xA

And,

xC

Take,

xA

And,

xB

xAB xAC

So,

xA

And,

xC

Hence, an element belongs to Bthen, it will belong to C, that is,

BC(1)

Now let,

xC

Hence,

xAC xAB

xA

And,

xB

Taking,

xA

And,

xC

xAC xAB

So,

xA

And,

xC

Hence, an element belongs to C then, it will belong to B, that is,

CB(2)

From equations (1) and (2),

B=C

Therefore,

B=C


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