Q.Let $A=\{5,6\}$; how many binary operations can be defined on this set?

Bijections of an $n$-set onto itself are permutations, so there are $n!$ of them.

A bijective function $f:A\to A$ with $|A| = n$ is just a permutation of the $n$ elements. The first element has $n$ choices of image, the next $n-1$, and so on, giving

$n\times(n-1)\times\cdots\times1 = n!.$ …

Total functions from $A$ to $B$ is $3^{2}=9$.

Each of the $2$ elements of $A$ can map to any of the $3$ elements of $B$ independently, s …

Number of one-one functions from $A$ to $B$ is $6$.

For an injective (one-one) function, the first element of $A$ has $3$ choices in $B$ and the second …

The number of functions from a set of size $m$ to a set of size $n$ is $n^m$; each of the $3$ elements of $B$ can map to either of $2$ elements of $G$.

For a function from $B$ to $G$, every element of $B$ must map to exactly one element of $G$.

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A binary operation is a map $A\times A\to A$; count $=|A|^{|A|^2}=2^{4}=16$.

A binary operation on $A$ assigns to each ordered pair in $A\times A$ an element of $A$.

With $A=\{5,6\}$, $|A|=2$ and $|A\times A|=2^2=4$.

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