If $A = \{a,b,c,d\}$ , then $|A\times A| = 16$ and there are $12$ ordered pairs in the form of $(x,y)$ where $x\neq y$.
From this how does the textbook get the answer $$4^4 \cdot 4^6 = \text{number of commutative closed binary operations on $A$}$$? I am very confused.
Is it because there are four choices for each of the assignments of x and y where x equals y(4) and 4 choices for the assignments of x and y where they dont equal esch other(6)?
A closed binary operation on $A$ is a function of the form $\ast: A\times A \rightarrow A$ (which I will write using infix notation).
If $\ast$ is commutative, then $x\ast y=y\ast x$, and thus $\ast$ is determined by exactly $10$ values: the elements of the form $(x,x)$, of which there are four, and half of the remaining elements (because of the commutativity), of which there are $12$, so $12/2=6$.
Thus, $\ast$ is determined by mapping the four elements of the form $(x,x)$ to the four values $a,b,c,d\in A$, giving rise to the value of $4^4$ as the number of such choices, and mapping the other six elements mentioned to the four values $a,b,c,d\in A$, giving rise to the value of $4^6$ as the number of these choices. Thus, the total number of such commutative closed binary operations on $A$ is given by $4^4\cdot 4^6$.
