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 =  =

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