Showing Associativity and Commutativity of a binary operation given by a Cayley table Let $*$ be a binary operation on the set $S:=\{0,1\}$ given by the following Cayley table:
\begin{array}{c|cc}
* & 0 & 1\\\hline
0 & 0 & 1\\
1 & 1 & 0
\end{array}
If I wish to show that this table is associative, how many different combinations of arguments do I have to show it for? Commutativity is easy to show because the table is symmetric along the main diagonal. However, I know since associativity and commutativity are universal properties, I am having trouble figuring out as stated: how many different combinations of arguments do I have to execute (e.g. $0*(0*1)=1*(0*0)$ to show $*$ is associative on the set $S$?

As an alternative to case-by-case checking (which does require $8$ checks), you could note that $a*b \equiv a+b \pmod{2}$.  Then you get that $*$ is associative because modular addition is associative.
