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.
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