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# If $\ast$ is a binary operation on the set $\{0, 1, 2, 3, 4, 5\}$ defined as $a \ast b = \left\{ \begin{array} {1 1} a+b, & \quad \text{ if a+b < 6} \\ a+b-6, & \quad \text{ if a+b \geq 6} \\ \end{array} \right.$ then draw the composition table for the operation and find the identity element and inverse of $4$ if exists.

$\begin{array}{1 1} \text{0 is identity and inverse of 4 is 2}\\ \text{identity and inverse do not exist} \\\text{0 is identity and inverse does not exist} \\ \text{1 is identity and inverse of 4 is 4} \end{array}$

Toolbox:
• An element $e \in N$ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
• The element $a \in X$ is invertible if there exist $b \in X$ such that $a*b=e=b*a$
Given the set $X=\{0,1,2,3,4,5\}$ where the binary operation $\ast$ is defined by $a * b= \left\{ \begin{array}{1 1} a+b & \quad if\;a+b < 6\\ a+b-6 & \quad if a+b \geq 6 \end{array} \right.$
The composition table is
$\textbf {Step 1: Checking if zero is the identity}$:
An element $e \in N$ is an identify element for operation * if $a*e=e*a$ for all $a \in N$
To check if zero is the identity, we see that $a*0=a+0=a \qquad for\;a \in x$ and also $0*a=0+a=a \qquad for \;a \in x$
Given $a \in X, \qquad a+0 < 6\;$ and also $\;0+a < 6$
$\Rightarrow 0$ is the identify element for the given given operation
From the table we can find the identity element
It is the leading element of the row which is same as first row.
$\textbf {Step 2: Finding the inverse of any element a}$:
The element $a \in X$ is invertible if there exist $b \in X$ such that $a*b=e=b*a$
In this case, $e=0 \rightarrow a*b=0=b*a$.
$\Rightarrow a*b = \left\{ \begin{array}{1 1} a+b=0=b+a & \quad if\;a+b < 6\\ a+b-6=0=b+a-6 & \quad a+b \geq 6 \end{array} \right.$
ie $a=-b \;or\; b=6-a$
but since $a,b \in X=\{0,1,2,3,4,5\}$, $\;a \neq -b$
Hence $b=6-a\;$ is the inverse of $a$, i.e., $a^{-1}=6-a, \;\forall a \in \{1,2,3,4,5\}$
Hence inverse element of $4$ is $2$
From the table we can find the inverse element of any element of A
That is In the table where ever identity element occur, the leading elements of the corresponding
row and column are inverse of each other
edited Feb 6, 2014