Let, A, B and C be the sets such that A∪B=A∪C and A∩B=A∩C.
To show: B=C
Let x∈B
⇒x∈A∪B (By def of union of sets)
⇒x∈A∪C (A∪B=A∪C)
⇒x∈A or x∈C
Case I
x∈A
Also, x∈B
∴x∈A∩B
⇒x∈A∩C (∵A∩B=A∩C)
∴x∈A and x∈C
∴x∈C
But x is an arbitrary element in B.
∴B⊂C .....(1)
Now, we will show that C⊂B.
Let y∈C
⇒y∈A∪C (by def of union of sets)
⇒y∈A∪B (A∪B=A∪C)
⇒y∈A or y∈B
Case I : When y∈A
Also, y∈C
⇒y∈A∩C
⇒y∈A∩B
⇒y∈A and y∈B
⇒y∈B
But y is an arbitrary element of C.
Hence, C⊂B .....(2)
From (1) and (2), w eget
∴B=C.