Question

prove that A intersection (B union C)=(A intersection B)union(A intersection C)​

Answer 1

Answer:

Let x in A ∩ (B U C)

Then x is in A and x is in (B U C).

If x is in B, then x is in A ∩ B.

If x is not in B, then x is in C, so x is in A ∩ C.

Thus x is in (A ∩ B) U (A ∩ C), and A ∩ (B U C) ⊆ (A ∩ B) U (A ∩ C).

Now assume x in (A ∩ B) U (A ∩ C) and similarly show that x is in A ∩ (B U C).

Then (A ∩ B) U (A ∩ C) ⊆ A ∩ (B U C).

So (A ∩ B) U (A ∩ C) = A ∩ (B U C)

Answer 2

Let x in A ∩ (B U C)

Then x is in A and x is in (B U C).

If x is in B, then x is in A ∩ B.

If x is not in B, then x is in C, so x is in A ∩ C.

Thus x is in (A ∩ B) U (A ∩ C), and A ∩ (B U C) ⊆ (A ∩ B) U (A ∩ C).

Now assume x in (A ∩ B) U (A ∩ C) and similarly show that x is in A ∩ (B U C).

Then (A ∩ B) U (A ∩ C) ⊆ A ∩ (B U C).

So (A ∩ B) U (A ∩ C) = A ∩ (B U C)