Union and Intersection

[toughcookie723]

I need some help with proving the following:

If A ∩ B=A U B, then A=B.

A ∩ B=A U B

By definition of intersection {x: x ∈ A and x ∈ B}
By definition of union {x: x ∈ A or x ∈ B}

It seems pretty obvious from the definitions for me but I don't think it's an adequate proof.

Could you please point me in the right direction.

Thanks!

 

[pka]

If A ∩ B=A U B, then A=B.

You must show that \(\displaystyle A\subseteq B\text{ and }B\subseteq A.\)

If \(\displaystyle x\in A\) then \(\displaystyle x\in A\cup B=A\cap B\) so \(\displaystyle x\in ?\).

Can you finish?

 

[toughcookie723]

Yes, I see what you mean, thank you so much!