A binary operation on the set <img alt="A= \left \{ 0,1,2,3,4,5 \right \}" src="https://entrancecorner.codecogs.com/gif.latex?A%3D%20%5Cleft%20%5C%7B%200%2C1%2C2%2C3%2C4%2C5%

A binary operation $\ast$ on the set $A= \left \{ 0,1,2,3,4,5 \right \}$ is defined as $a\ast b= \left\{\begin{matrix} a+b, \, if\, a+b< 6 & \\ a+b-6,\, if\, a+b\geq 6& \end{matrix}\right.$
Write the operation table for $a\ast b\, in\, A\cdot$ Show that zero is the identity for this operation $\ast$ and each elemnt $'a'\neq 0$ of the set is invertible with 6-a, being the inverse of 'a'.

Answers (1)

we ahve $a\ast b= \left\{\begin{matrix} a+b\, \ if \ \, a+b< 6 & \\ a+b-6\, \ if \ \ \, a+b\geq 6 & \end{matrix}\right.$ defined on the set A
$A= \left \{ 0,1,2,3,4,5 \right \}$
operation table of the binary operation $\ast$ is given below

$\ast$	0	1	2	3	4	5
0	0	1	2	3	4	5
1	1	2	3	4	5	0
2	2	3	4	5	0	1
3	3	4	5	0	1	2
4	4	5	0	1	2	3
5	5	0	1	2	3	4

Let x be the identity for element a
so, $a\ast\, x= a\, \ if \ \: a+x< 6\: \ then \ \, a+x= a\; \ \ or\ \; \ if \ \; a+x\geq 6\; \ then,\; a+x-6= a\cdot$
ie $a+x< 6\; then\; x= 0\epsilon A\; or\; a+x\geq 6\; then\; x= 6\not{\epsilon }A$
$\therefore x= 0$ is the identity element for this operation.
Also let y be the innverse of each non-zero element a.
Then $a\ast y= 0$
If $a+y< 6$ then $a+y= 0$ or $if\, a+y\geq 6$ then $a+y- 6= 0$
ie $a+y< 6$ then $y= a\not{\varepsilon }A$ for all $a\, \varepsilon \, A-\left \{ 0 \right \}$
or $a+y\geq 6$ then $y= 6-a\, \varepsilon \, A$ for all $a\, \varepsilon \, A-\left \{ 0 \right \}$
$\therefore y= 6-a$ is the inverse of each non-zero element
'a' of A