Can Set Theory Prove Equality and Intersection Properties?

[brad sue]

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?

Thank you for your help
B

 

[Hurkyl]

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

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

but that said...

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

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

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

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

=A I (~B U A)

This step is wrong too.

how can I treat this problem?

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

 

[e(ho0n3]

You should use Venn diagrams: http://en.wikipedia.org/wiki/Venn_diagram. By the way, why did you post this in the Calculus forum?

 

[Hurkyl]

Because it's the Calculus & Beyond forum.