Determine whether each of the following operations define a binary ope

1.	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.
Answer» 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,b∈S Since a∈S and b∈S, 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, ⇒ ax₆b∈S ...... (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.*

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.

Answer»

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,b∈S

Since a∈S and b∈S,

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,

⇒ ax₆b∈S ...... (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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