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Q. 14 Define a binary operation ∗ on the set $\{0, 1, 2, 3, 4, 5\}$ as

$a * b = \left\{\begin{matrix} a + b &if\;a+b < 6 \\ a+ b -6 & if\;a+b\geq6 \end{matrix}\right.$

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

Answers (1)

best_answer

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

$a * b = \left\{\begin{matrix} a + b &if\;a+b < 6 \\ a+ b -6 & if\;a+b\geq6 \end{matrix}\right.$

An element $c \in X$ is identity element for operation *, if $a*c=a=c*a \, \, \forall \, \, a \in X$

For $a \in X$ ,

$a *0 = a+0=a\, \, \, \, \, \, \, \, \, \, \, \, \left [ a \in X \Rightarrow a+0< 6 \right ]$

$0 *a = 0+a=a\, \, \, \, \, \, \, \, \, \, \, \, \left [ a \in X \Rightarrow a+0< 6 \right ]$

$\therefore \, \, \, \, \, \, \, \, \, a*0=a=0 *a$ $\forall a \in X$

Hence, 0 is identity element of operation *.

An element $a \in X$ is invertible if there exists $b \in X$ ,

such that $a*b=0=b*a$ i.e. $\left\{\begin{matrix} a + b =0=b+a &if\;a+b < 6 \\ a+ b -6=0=b+a-6 & if\;a+b\geq6\end{matrix}\right.$

means $a=-b$ or $b=6-a$

But since we have X = $\{0, 1, 2, 3, 4, 5\}$ and $a,b \in X$ . Then $a\neq -b$ .

$\therefore b=a-x$ is inverse of a for $a \in X$ .

Hence, inverse of element $a \in X$ , $a\neq 0$ is 6-a i.e. , $a^{-1} = 6-a$

Posted by

seema garhwal

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