Question

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

Answer 1

Let A, B and C be the sets such that A∪B=A∪CandA∩B=A∩C.
To show: B = C
Let x∈B
⇒x∈A∪B[B⊂A∪B]⇒x∈A∪C[A∪B=A∪C]⇒x∈Aorx∈CCaseI×∈
A
Also, x∈B
x∈A∩B
∴
⇒x∈A∩C[∵A∩B=A∩C]∴x∈Aandx∈C∴x∈C∴B⊂C
Similarly, we can show that C⊂B.
∴B=C