Question

Prove A ∩ (A' U B) = A ∩ B ?


Note: It's A complement
Any brainly stars ⭐✨.please help​

Answer 1

SOLUTION :

❒ To Prove :-

• A∩(A′ U B) = A∩B

We know that,

If U is an universal set, then A∩A′ = ϕ

❒ Proof :-

★ L.H.S. = A ∩ (A′ U B)

= (A∩A′) U (A∩B)

= ϕ U (A∩B)

= A∩B = R.H.S.

∴ A∩(A′ U B) = A∩B ★

• Hence, Proved.

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This can be proved with the help of an example.

❒ To Prove :-

• A∩(A′ U B) = A∩B

Suppose,

U is an universal set where A and B are two different sets under U. [Referred to the Venn Diagram]

Where,

• A = {0, 2, 4, 6, 8, 10}

• B = {1, 2, 3, 4, 5, 6, 7, 8}

Here,

• A′ U B = {1, 3, 5, 7} U {1, 2, 3, 4, 5, 6, 7, 8} = {1, 2, 3, 4, 5, 6, 7, 8}

• A∩B = {0, 2, 4, 6, 8, 10} ∩ {1, 2, 3, 4, 5, 6, 7, 8} = {2, 4, 6, 8}

Now,

• A∩(A′ U B) = {0, 2, 4, 6, 8, 10} ∩ {1, 2, 3, 4, 5, 6, 7, 8}

➜ A∩(A′ U B) = {2, 4, 6, 8}

• A∩B = {2, 4, 6, 8}

∴ A∩(A′ U B) = A∩B ★

• Hence, Proved.