Determine whether the following operations define a binary operat… - Vidyadip

Question

Determine whether the following operations define a binary operation on the given set or not:
$'\times _6'$ on $S = \{1, 2, 3, 4, 5\}$ defined by, $a \times _6 b =$ Remainder when ab is divided by $6.$

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Answer

Consider the composition table,

$\times _6$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$2$	$3$	$4$	$5$
$2$	$2$	$4$	$0$	$2$	$4$
$3$	$3$	$0$	$3$	$0$	$3$
$4$	$4$	$2$	$0$	$4$	$2$
$5$	$5$	$4$	$3$	$2$	$1$

Here all the elements of the table are not in $S.$
For $a =2$ and $b =3$,
$\text{a}\times_6\text{b}= 2 \times_63=$ remainder when $6$ divided by $6=0\neq\text{S}$
Thus, $\times _6$ is not a binary operation on $S.$

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