Showing Associativity and Commutativity of a binary operation given by a Cayley table

Question

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 $S$?

Answer 1

Associativity means that for all $a, b $ and $c$ in $S$, $(a * b) * c = a * (b * c).$

Since $S$ has only two elements,

there are only two possibilities to check for $a$, two for $b$, and two for $c$;

altogether, there are $2\times2\times2=8$ possibilities to check:

$(0*0)*0=0*(0*0)$

$(0*0)*1=0*(0*1)$

$(0*1)*0=0*(1*0)$

$(0*1)*1=0*(1*1)$

$(1*0)*0=1*(0*0)$

$(1*0)*1=1*(0*1)$

$(1*1)*0=1*(1*0)$ and

$(1*1)*1=1*(1*1)$.