Determine whether each of the following operations define a binary operation on the given set or not:
‘+6’ on S = {0,1,2,3,4,5} defined by a +6 b= 
Given that ‘*’ is an operation that is valid for the numbers in the set S = {0,1,2,3,4,5} and it is defined as given:
⇒ a +6 b = 
, 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 the number that present in the set ‘S’ as a result of the operation,
⇒ a + 6b∈S ...... (1)
Since the addition of any two whole numbers must give a whole number, we can say that,
⇒ a + b≥0
For a + b<6, a + b∈S
⇒ a + b≥6 ⇒ a + b – 6≥0∈W
But according to the numbers given in the problem, the maximum value of the sum we can attain is 6.
So, we can say that a + b – 6∈S.
∴ The operation ‘ + 6’ defines a binary operation on S.