# Proof in Discrete math

1. Sep 9, 2006

Hi, I would like some help for the following problems.

please bear with me with my special notation:
I- intersection
U- union
S- universal set
~- complement

I need to prove that: let be A and B two sets. prove
(A U B) I (A I (~B))=A

what I did is:
(A U B) I (A I (~B))

=[(A I B) U A] I [(A I B) U ~B]/distribution

=A I [(~B U A) I (~B U B)]

=A I [(~B U A) U S)

=A I (~B U A)

=(A I (~B)) U (A I A)

=AUA=A //i'm not sure here (A I (~B)) =A

problem 2

Can we conclude that A=B if A,B,C are sets such that
i) A U C = B U C
ii) A I C = B I C

how can I treat this problem?

B

2. Sep 9, 2006

### Hurkyl

Staff Emeritus
As stated, that's not true. (e.g. let A be any nonempty set, and let B equal A)

but that said...

It doesn't look like you applied this rule right.

This step is wrong too.

I would try and draw a picture to help with my intuition.

3. Sep 9, 2006

4. Sep 9, 2006

### Hurkyl

Staff Emeritus
Because it's the Calculus & Beyond forum.