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:
'×₆' on S = {1, 2, 3, 4, 5} defined by, a ×₆ b = Remainder when ab is divided by 6.

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Answer

Consider the composition table,

×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, ×₆ is not a binary operation on S.

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