Question

Defined a binary operation $\ast$ on the set $\{0,1,2,3,4,5\}$ as
$a\ast b=\{\begin{array}{cc}a+b,& ifa+b<6\\ a+b-6& ifa+b\ge 6\end{array}$
Show that zero is the identity for this operation and each element $a\ne 0$ of the set is invertible with $6-a$ being the inverse of $a$.

Answer 1

$0\ast 0=0+0=0$

$0\ast 1=0+1=1$

$0\ast 2=0+2=2$

$0\ast 3=0+3=3$

$0\ast 4=0+4=4$

$0\ast 5=0+5=5$

i.e $0\ast a=0+a=a$

and $a\ast 0$$=$$a+0$$=$$a$

$\Rightarrow$$a\ast 0=0\ast a=a$ $\forall \text{a}\in \text{{ 0,1,2,3,4,5}}$

$\Rightarrow$ $0$ is the identity for this operation

Let $\text{a}\in \text{{ 0,1,2,3,4,5}}$ and $b=6-a$

Then $a\ast b=a+b-6$

$=a+6-a-6$

$=0$ (since $a+b=6$)

and $b\ast a=b+a-6$

$=6-a+a-6$ (since $a+b=6$)

$\Rightarrow$Each element $\text{a}\in \text{{ 0,1,2,3,4,5}}$ is invertible and has $6-a$ as its inverse