# Set theory proof

1. Feb 3, 2009

### Blue_Wind

1. The problem statement, all variables and given/known data
A,B and C are sets.
Prove (A∩B)C = AC∩BC is FALSE
That is, I have to give a counterargument for this statement.

2. Relevant equations
I can't find a counterargument directly. My professor suggest trying to prove the statement to find a problem and come up with the counterargument.
To prove this is false, first must prove that
AC∩BC$$\subseteq$$(A∩B)C is false, OR
(A∩B)C$$\subseteq$$AC∩BC is false.

3. The attempt at a solution
I have proven (A∩B)C$$\subseteq$$AC∩BC is true by:
• w is a string
• Let w$$\in$$(A∩B)C then $$\exists$$u$$\in$$(A∩B)and $$\exists$$v$$\in$$C where w=uv
• If $$\exists$$u$$\in$$A then w=uv$$\in$$AC and $$\exists$$u$$\in$$B and w=uv$$\in$$BC
• Hence (A∩B)C$$\subseteq$$AC∩BC

However, I wasn't able to prove AC∩BC$$\subseteq$$(A∩B)C is false.

• w is a string
• Let w$$\in$$AC and w$$\in$$BC
• Then $$\exists$$u$$\in$$A and $$\exists$$v$$\in$$C where w=uv
• Also $$\exists$$u$$\in$$B and $$\exists$$v$$\in$$C where w=uv
• Then $$\exists$$u$$\in$$A∩B and $$\exists$$v$$\in$$C
• Hence AC∩BC$$\subseteq$$(A∩B)C ?

My professor said that the second part is wrong, but I have already tried over an hour but still can not make the second part false nor just come up with a counterargument.

I'm really not good with logic, can anyone help me?
I still have a lot of programming assignment waiting for me to do.

2. Feb 7, 2009

### nitro

you don't really need to prove anything..
you just have to come up with a counterexample

If you draw a Venn diagram of the 3 sets you can construct your counter example

Good luck!