Question

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

Answer 1

Answer:

B=C

Step-by-step explanation:

AUB =AUC

Hence we know

element of A + element of B = element of A +element of C

element of B =Element of C

Hence B =C

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Answer 2

Answer:

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Step-by-step explanation:

Step-1:Prove using suitable formula of sets.

It is given that,

A∪B = A∪C …(1)

A∩B = A∩C…(2)

Taking  ’∩ C’ on both sides in equation (1)

(A∪B)∩C = (A∪C)∩C

We know that,

(A∪B)∩C = (A∩C)∪(B∩C) and (A∪C)∩C = C

So,

(A∩C)∪(B∩C)=C

(A∩B)∪(B∩C)=C…(3)[From(2))

Again,

Taking  ’∩ B’ on both side in equation (1)

(A∪B)∩B = (A∪C)∩B

B = (A∩B)∪(C∩B)

B = (A∩B)∪(B∩C)

B = C[From (3)]

Hence, proved.