Question

Use tables to show that ${\mathit{Z}}_{\mathbf{2}}\mathbf{\times }{\mathit{Z}}_{\mathbf{3}}$ is isomorphic to ring$\mathit{R}$ of Exercise 2 in Section 3.1.

Short answer

Hence proved that$Z_2\times Z_3$ is isomorphic to ring$R$

Step-by-step solution

Use tables to show that Z2×Z3   is isomorphic to ring R  of Exercise 2 in Section 3.1.

If any ring R has elements such that,$a,b\in R$, then, addition and multiplication of the function of its elements is respectively given by:

$$\begin{gathered} f\left(a+b\right)=f\left(a\right)+f\left(b\right) \\ f\left(ab\right)=f\left(a\right)f\left(b\right) \end{gathered}$$

Addition and Multiplication Table

The given table in exercise 2 in 3.1 is:

+	0	e	b	c
0	0	e	b	c
e	e	0	c	b
b	b	c	0	e
c	c	b	e	0

$·$	0	e	b	c
0	0	0	0	0
e	0	e	b	c
b	0	b	b	0
c	0	c	0	c

The addition and multiplication table in are:

+	(0,0)	(1,1)	(1,0)	(0,1)
(0,0)	(0,0)	(1,1)	(1,0)	(0,1)
(1,1)	(1,1)	(0,0)	(0,1)	(1,0)
(1,0)	(1,0)	(0,1)	(0,0)	(1,1)
(0,1)	(0,1)	(1,0)	(1,1)	(0,0)

.	(0,0)	(1,1)	(1,0)	(0,1)
(0,0)	(0,0)	(0,0)	(0,0)	(0,0)
(1,1)	(0,0)	(1,1)	(1,0)	(0,1)
(1,0)	(0,0)	(1,0)	(1,0)	(0,0)
(0,1)	(0,0)	(0,1)	(0,0)	(0,1)

Clearly, if we have a mapping as$f:Z_6\to Z_2\times Z_3$:

Then, we get:$0\to \left(0,0\right),e\to \left(1,1\right),b\to \left(1,0\right),c\to \left(0,1\right)$

The function is an isomorphic function. Hence proved, $Z_2\times Z_3$ is isomorphic to ring R.