# Prove A = (A\B) ∪ (A ∩ B)

I have to demonstrate this formulae:

Prove A = (A\B) ∪ (A ∩ B)

But it seems to me that it is false.

(A\B) ∪ (A ∩ B)

• X ∈ A/B => { x ∈ A and x ∉ B }

or

• X ∈ A ∩ B => { x ∈ A and x ∈ B }

so:

x ∈ A ∩ B

so: A ≠ (A\B) ∪ (A ∩ B)

Did I solve the problem or I am just blind?

-
Think of $(A\backslash B)$ as $A \cap B^c$. –  Rudy the Reindeer Jan 18 '11 at 18:07

-