Chapter 3.3B, Problem 33A

To determine

a.

Whether the set $S$ is closed with respect to operation $\odot$, where operation $\odot$ is defined on the set $S=\left\{a,b,c\right\}$, as shown in the following table.

To determine

b.

Whether operation $\odot$ is commutative on $S$, where operation $\odot$ is defined on the set $S=\left\{a,b,c\right\}$, as shown in the following table.

To determine

c.

Whether there is an identity for $\odot$ on $S$, where operation $\odot$ is defined on the set $S=\left\{a,b,c\right\}$, as shown in the following table. If yes, what is the identity?

To determine

d.

Whether operation $\odot$ is associative on $S$, by trying several examples, where operation $\odot$ is defined on the set $S=\left\{a,b,c\right\}$, as shown in the following table.