what is the number of binary operations on the set {a,b} and how?

 

The number of functions from a set with m elements to a set with n elements is n^m.

 Binary operation is also a function from a set Sx S  to S.

 So, The number of binary operations on the set S  with n elements is (nⁿ)^2

So , the number of binary operations on the set {a,b} with 2 elements is( 2²)^2  = 2 ⁴