Let A, B, and C be the sets such that AUB=AUC and AnB=An C. Show
that B =C
Answers
Answer:
Given that AUB = AUC
⇒ (AUB) ∩ C = (AUC) ∩C
⇒ (A∩C) U (B∩C) = C [ ∴(AUC)∩C = C ]
⇒ (A∩B) U (B∩C) = C ..........(1) [ ∴(A∩C) = A∩B ]
Again AUB = AUC
(AUB) ∩ B = (AUC) ∩ B
B = (A∩B) U (C∩B)
= (A∩B) U (B∩C) ...........(2)
From 1 & 2 we get
B = C
mrk me brinliest plzzz
Answer:
mark me as brainliest and support me to give more valuable answers.
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.