Define a binary operation $*$ on the set {0, 1, 2, 3, 4, 5} as $a * b = \left\{ {\begin{array}{llllllllllllllllllll}{a + b,}&{if}&{a + b < 6}\\{a + b - 6,}&{if}&{a + b \ge 6}\end{array}} \right.$

Show that zero is the identity forth is operation and each element$a \ne 0$ of the set is invertible with $6 - a$ being the inverse of a.

For identity

If e be the identity element, then a $*$ e $=$ e $*$ a $=$ a

Now, a $*$ 0 $=$ a + 0 $=$ a and 0 $*$ a $=$ 0 + a $=$ a

Thus, a $*$ 0 $=$ 0 $*$ a $=$ a. Hence, 0 is the identity element of the operation.

For inverse

If b be the inverse of a. then a $*$ b $=$ b $*$ a $=$ e.

Now a $*$ (6$-$ a) $=$ a + (6 $-$ a) $-$ 6 $=$ 0

and (6 $-$ a ) $*$ a $=$ (6 $-$ a) + a $-$ 6 $=$ 0.

Hence, each element a of the set is invertible with inverse 6$-$a.