Question

Let $A=\{1,-1, i,-i\}$, where $i=\sqrt{-1}$. Draw the composition table corresponding to binary operation 'multiplication' on A.

Answer 1

We have

$1 \times 1=1$

$(-1) \times 1=-1$

$\mathrm{i} \times 1=\mathrm{i}$

$(-1) \times \mathrm{i}=-\mathrm{i} \\ $

$1 \times(-1)=-1$

$(-1) \times(-1)=1$

$\mathrm{i} \times(-1)=-\mathrm{i}$

$(-i) \times(-1)=i \\$

$1 \times \mathrm{i}=\mathrm{i}$

$(-1) \times \mathrm{i}=-\mathrm{i}$

$\mathrm{i} \times \mathrm{i}=-1$

$(-i) \times i=1 \\$

$1 \times \mathrm{(-i)}=\mathrm{-i}$

$(-1) \times \mathrm{(-i)}=-\mathrm{i}$

$\mathrm{i} \times \mathrm{(-i)}=1$

$\mathrm{(-i)} \times \mathrm{(-i)}=-1 $

The required composition table is given below :

x	1	-1	i	-i
1	1	-1	i	-i
-1	-1	1	-i	i
i	i	-i	-1	1
-i	-i	i	1	-1