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

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

Answer 2

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.

Answer 3

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.