# Let X be a non empty set n distinct elements. the total number of binary operations that can be defined on X is

Solution:

Given that n(X) = n , By definition , a binary opretion is a mapping

$\\ o:X\times X\rightarrow X,$Defined by $\\ (x,y)\rightarrow xoy$

Now, $\\ n(X\times X)=n^2$

for each ordered pair $\\ (x,y)\in X\times X$, there are n choices .

Therefore , total number of binary operations=n^2

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