Help with a Union Problem

[MattF]

Assume A,B, and C are finite sets. Give, with proof, formulas for |A union B| and |A union B union C|.

What is this really asking for? A union just means that it is in either A or B correct? I am hung up on what a formula would be. I have ran across the inclusion - exclusion principle which makes sense for the |A union B| case ( |A union B| = |A|+|B| -|A int B| ) but what would be a proof for this?

Thank you.

 

[pka]

Assume A,B, and C are finite sets. Give, with proof, formulas for |A union B| and |A union B union C|.
inclusion - exclusion principle which makes sense for the |A union B| case ( |A union B| = |A|+|B| -|A int B| ) but what would be a proof for this?

\(\displaystyle |A|+|B|\) counts the number of elements in \(\displaystyle A\) plus the number of elements in \(\displaystyle B\).
But \(\displaystyle |A\cup B|\) is the number of elements in \(\displaystyle A\text{ or }B\).
Thus we have may counted some elements in \(\displaystyle A\cap B\) twice.
So subtract it them off.

\(\displaystyle |A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|\)