Chapter 3.2, Problem 15MC

To determine

a.

Whether the system is closed with explanation where, a mathematical system uses only four symbols, $a,b,c$, and $d$, and the operation $\Delta$ as shown in the table,

$\begin{array}{|ccccc|}\hline \Delta & a& b& c& d\\ a& a& b& c& d\\ b& b& c& d& a\\ c& c& d& a& b\\ d& d& a& b& c\\ \hline\end{array}$

To determine

b.

Whether the operation $\Delta$ is commutative with explanation where, a mathematical system uses only four symbols, $a,b,c$, and $d$, and the operation $\Delta$ as shown in the table,

$\begin{array}{|ccccc|}\hline \Delta & a& b& c& d\\ a& a& b& c& d\\ b& b& c& d& a\\ c& c& d& a& b\\ d& d& a& b& c\\ \hline\end{array}$

To determine

c.

Whether the operation $\Delta$ have any identity where, a mathematical system uses only four symbols, $a,b,c$, and $d$, and the operation $\Delta$ as shown in the table,

$\begin{array}{|ccccc|}\hline \Delta & a& b& c& d\\ a& a& b& c& d\\ b& b& c& d& a\\ c& c& d& a& b\\ d& d& a& b& c\\ \hline\end{array}$

To determine

d.

Whether the operation $\Delta$ is associative by using several examples where, a mathematical system uses only four symbols, $a,b,c$, and $d$, and the operation $\Delta$ as shown in the table,

$\begin{array}{|ccccc|}\hline \Delta & a& b& c& d\\ a& a& b& c& d\\ b& b& c& d& a\\ c& c& d& a& b\\ d& d& a& b& c\\ \hline\end{array}$