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$$\subseteq$$AC∩BC is false.

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.

 

[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!

 

[Mene]

Counterargument:

The statement (A∩B)C=AC∩BC is not always false. In fact, it can be true in certain cases. For example, let A={1,2,3}, B={2,3,4}, and C={3,4,5}. Then (A∩B)C={(1,2,3), (2,3,4), (3,4,5)} and AC∩BC={(1,2,3), (2,3,4), (3,4,5)}. In this case, the statement is true. Therefore, the statement cannot be proven to be always false and a counterargument cannot be found.