Question

A binary operation * on the set {0,1,2,3,4,5} is defined as
$a\ast b=\left\{\begin{array}{c}a+b,ifa+b<6\\ a+b-6ifa+b\ge 6\end{array}\right\}$ show that zero is the identity element of this operational each element 'a' of the set is invertible with 6-a being the inverse of 'a'

Answer 1

Given $X=\{0,1,2,3,4,5\}$

$a\ast b=\left\{\begin{array}{c}a+b,ifa+b<6\\ a+b-6ifa+b\ge 6\end{array}\right\}$

To check if zero is the identify, we see that $a\ast 0=a+0=a$

for all $a\in x$ and also $0\ast a=0+a=a$ for $a\in x$

Given $a\in X,a+0<6$ and also $0+a<6$

$\Rightarrow 0$ is the identity element for the given operation.

Now

The element $a\in X$ is invertible if there exist $b\in x$ such that $a\times b=e=b\ast a$

In this case, $e=0\to a\ast b=0=b\ast a$

$\Rightarrow a\ast b=\left\{\begin{array}{c}a+b=0=b+a,ifa+b<6\\ a+b-6=0=b+a-6ifa+b\ge 6\end{array}\right\}$

i.e, $a=-b$ or $b=6-a$

but since, $ab\in x=\{0,1,2,3,4,5\}a\ne -b$

Hence $b=6-a$ is the inverse of $a$ i.e, $a^{-1}=6-a$

$\forall a\in \{1,2,3,4,5\}$