- #1

Blue_Wind

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## 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[tex]\subseteq[/tex](A∩B)C is false, OR

(A∩B)C[tex]\subseteq[/tex]AC∩BC is false.

## The Attempt at a Solution

I have proven (A∩B)C[tex]\subseteq[/tex]AC∩BC is true by:

- w is a string
- Let w[tex]\in[/tex](A∩B)C then [tex]\exists[/tex]u[tex]\in[/tex](A∩B)and [tex]\exists[/tex]v[tex]\in[/tex]C where w=uv
- If [tex]\exists[/tex]u[tex]\in[/tex]A then w=uv[tex]\in[/tex]AC and [tex]\exists[/tex]u[tex]\in[/tex]B and w=uv[tex]\in[/tex]BC
- Hence (A∩B)C[tex]\subseteq[/tex]AC∩BC

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

- w is a string
- Let w[tex]\in[/tex]AC and w[tex]\in[/tex]BC
- Then [tex]\exists[/tex]u[tex]\in[/tex]A and [tex]\exists[/tex]v[tex]\in[/tex]C where w=uv
- Also [tex]\exists[/tex]u[tex]\in[/tex]B and [tex]\exists[/tex]v[tex]\in[/tex]C where w=uv
- Then [tex]\exists[/tex]u[tex]\in[/tex]A∩B and [tex]\exists[/tex]v[tex]\in[/tex]C
- Hence AC∩BC[tex]\subseteq[/tex](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.