Let A = {1, 2}, how many binary operations can be defined on this set?

The correct option is 16.

To understand why this is the case, let's break down the definition and calculation of binary operations on a set step-by-step.

Step 1: What is a binary operation?
A binary operation on a non-empty set $A$ is a function that takes two elements from $A$ and maps them to a single element in $A$. Mathematically, a binary operation is a function of the form:
$f:A\times A\to A$

Step 2: Determine the size of the domain
The domain of our function is the Cartesian product $A\times A$. Given that the set $A=\{1,2\}$ has $n=2$ elements, the Cartesian product $A\times A$ contains all possible ordered pairs of elements from $A$:
$A\times A=\left\{\right(1,1),(1,2),(2,1),(2,2\left)\right\}$

Thus, the number of elements in the domain is:
$|A\times A|=n^2=2^2=4$

Step 3: Calculate the total number of functions
For any two finite sets $X$ and $Y$, the total number of functions from $X$ to $Y$ is given by ${\left|Y\right|}^{\left|X\right|}$ (the number of choices in the codomain raised to the power of the number of inputs in the domain).

Here, the domain $X=A\times A$ has 4 elements, and the codomain $Y=A$ has 2 elements. Therefore, the number of possible binary operations is:
$2^4=16$

In general, for a set with $n$ elements, the total number of binary operations is:
$n^{\left(n^2\right)}$

For $n=2$, this formula yields:
$2^{\left(2^2\right)}=2^4=16$