Question

Let A , B And C be The Sets Such That A U C & A Π B = A Π C Show That B = C​

Answer 1

Answer:

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Answer 2

$\mathfrak{\huge{\blue{\underline{\underline{AnswEr :}}}}}$

» Given :

A, B and C are Three Sets Such that -

A ∪ C = A ∩ B and A ∪ C = A ∩ C

» To Prove :

⚘ B = C

» Proof :

⇒ A∪B = A∪C

⇒ (A∪B) ∩C = (A∪C) ∩C

⇒ (A∩C) ∪ (B∩C) = C

⠀⠀ [ ∴(A∪C)∩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 $\mathscr\green{we\:get\:\:\:B\:=\:C}</p><p>$