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 $S$?

An exhaustive research of all the triples over a set with $2$ elements requires to verify $2^3=8$ different equations.
Instead, we may try with the method shown in Verification of Associativity of a Binary Operation by S. Kamal Abdali. From the Cayley table
\begin{array}{c|cc}
* & 0 & 1\\\hline
0 & 0 & 1\\
1 & 1 & 0
\end{array}
we get:
\begin{array}{c|cc}
* & 0 & 1\\\hline
0 & 0 & 1\\
1 & 1 & 0\\\hline
0 & 0 & 1\\\hline
\end{array}
then
\begin{array}{c|cc}
* & 0 & 1\\\hline
0 &   &  \\
1 &   & 0\\\hline
0 & 0 & 1\\\hline
\end{array}
and
\begin{array}{c|cc}
* & 0 & 1\\\hline
0 &   &  \\
1 &   &  \\\hline
0 &   & 1\\\hline
\end{array}
So, pretty quickly, we verified this operation to be associative.
