Determine whether each of the following operations define a binary operation on the given set or not:

‘x6’ on S={1,2,3,4,5} defined by a x6 b = Remainder when ab is divided by 6.

Given that ‘x6’ is an operation that is valid for the numbers in the Set S = {1,2,3,4,5} and it is defined as given:


ax6b = Remainder when ab is divided by 6, where a,bS


Since aS and bS,


According to the problem it is given that on applying the operation ‘*’ for two given numbers in the set ‘S’ it gives one of the numbers in the set ‘S’ as a result of the operation,


ax6bS ...... (1)


Let us take the values of a = 3, b = 4,


ab = 3×4


ab = 12


We know that 12 is a multiple of 6. So, on dividing 12 with 6 we get 0 as remainder which is not in the given set ‘S’.


The operation ‘x6’ does not define a binary operation on set S.


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