I have been given set $M=\{0,1,2,3\}$ and a binary operation $a\circ b=\max\{a,b\}$. I need to prove that this set is a monoid. So In order to prove that I need to prove that $M$ is associative under that operation. How am I supposed to prove it? Is it fine if I take $3$ elements and just prove that it works for that set, or do I need to prove for each possible element selection?
How to prove associativity
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0[Max and Min are Associative](http://www.proofwiki.org/wiki/Max_and_Min_are_Associative) - ProofWiki – 2012-10-28