how to calculate the number of binary operations on any set A , say of 4 elements?
Let S be a finite set having n elements.
Then S × S has n2 elements.
Since a binary operation on S is a function from S × S to S.
∴ Total number of binary operations on S is equal to the number of functions from S × S to S.
Also, cardinality of domain (i.e. S × S) is n2 and cardinality of co-domain (i.e. S) is n.
So, total number of functions from S × S to S are
∴ Total number of binary operations on set S having n elements is .
Now, you are saying that on any set A (i.e. S is given to be A here) having 4 elements (i.e. value of n is 4)
Using the formula derived above, The number of binary operations on
Set A = =