Prove (A∩B)C=AC∩BC is FALSE: Counterargument Needed

[Blue_Wind]

Homework Statement

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.

Homework 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\subseteqAC∩BC is false.

The Attempt at a Solution

I have proven (A∩B)C\subseteqAC∩BC is true by:

• w is a string
• Let w\in(A∩B)C then \existsu\in(A∩B)and \existsv\inC where w=uv
• If \existsu\inA then w=uv\inAC and \existsu\inB and w=uv\inBC
• Hence (A∩B)C\subseteqAC∩BC

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

• w is a string
• Let w\inAC and w\inBC
• Then \existsu\inA and \existsv\inC where w=uv
• Also \existsu\inB and \existsv\inC where w=uv
• Then \existsu\inA∩B and \existsv\inC
• 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.

 

[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!