Question

Determine whether the following operation 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.

Answer 1

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,

a ×₆ b = 2 ×₆ 3 = remainder when 6 divided by 6 = 0 ≠ S

Thus, ×₆ is not a binary operation on S.