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 *n*^{2 }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 *n*^{2 }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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