HW2 Problem 3

Complete the table below so as to define a commutative operation $*$ on $S=\{a,b,c,d\}$.

(1)
\begin{array} {c||c|c|c|c} * & a & b & c & d \\ \hline a & a & b & c & \\ \hline b & b & d & & c \\ \hline c & c & a & d & b \\ \hline d & d & & & a \\ \end{array}

Solution:

A table defining a commutative operation is reflected across the diagonal such that $a*d=d*a$, $b*c=c*b$, $d*b=b*d$, and $d*c=c*d$.

(2)
\begin{array} {c||c|c|c|c} * & a & b & c & d \\ \hline a & a & b & c & \color{red}d \\ \hline b & b & d & \color{red}a & c \\ \hline c & c & a & d & b \\ \hline d & d & \color{red}c & \color{red}b & a \\ \end{array}