Chapter 6.1, Problem 3E

a.

To determine

To find the identity element of the provided system.

b.

To determine

To check the operation is associative or not.

c.

To determine

To check the operation is commutative or not.

d.

To determine

To find the inverse element of each.

e.

To determine

To check that ${\mathrm{B}}_1=\{\mathrm{a},\mathrm{b},\mathrm{c}\}$ is closed or not under the operation.

f.

To determine

To check that ${\mathrm{B}}_2=\{\mathrm{a},\mathrm{c}\}$ is closed or not under the operation

g.

To determine

To find all the subsets of that are closed under the operation.

h.

To determine

To verify whether the provided expression $\mathrm{x}\circ \mathrm{y}=\mathrm{y}\circ \mathrm{x}$ is true or not.