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Question

Let A,B, and C be the sets such that AB=AC and AB=AC. Show that B=C

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Solution

Let, A, B and C be the sets such that AB=AC and AB=AC.
To show: B=C
Let xB
xAB (By def of union of sets)
xAC (AB=AC)
xA or xC
Case I
xA
Also, xB
xAB
xAC (AB=AC)
xA and xC
xC
But x is an arbitrary element in B.
BC .....(1)
Now, we will show that CB.
Let yC
yAC (by def of union of sets)
yAB (AB=AC)
yA or yB
Case I : When yA
Also, yC
yAC
yAB
yA and yB
yB
But y is an arbitrary element of C.
Hence, CB .....(2)
From (1) and (2), w eget
B=C.

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