For any two finite sets $S$, $T$ show that the average of $|S|$and $|T|$ does not exceed $|S \cup T|$

Question

I have this problem for my mathematics course and I am not exactly sure how to approach it. If I could just get some hints about how to start this problem it would be greatly appreciated.

I have some ideas about how I should start:

I was thinking I could do $(|S|+|T|)/2$ and have its less than or equal to $|S|+|T|-|S \cap T|$. I got this far but I am not sure what to do after this.

Thanks in advance :)

Answer 1

Inclusion/exclusion gives you $$|S|+|T|=|S\cup T|+|S\cap T|\leq 2|S\cup T|\ .$$