Question

The number of binary operations on the set $\{1,2,3\}$ is _________.

• A. $3^9$
• B. $9^3$
• C. $27$
• D. $3!$

Answer 1

The correct option is A $3^9$

Let us denote this set by $S$, then $|S|=3$.

A binary relation defined on the elements of $S$ maps all elements in $S\times S$ to elements in $S$ by definition.

In this case any binary relation will thus have $3^2=9$ inputs each of which is an ordered pair of elements from $S$ and only $3$ number of possible outputs.

If all possible binary operations are considered then it is possible to assign any of the $3$ outputs to any of the $9$ inputs. So the number of all binary operations would exactly be $3^9$.

So option A is the correct answer.