MCQ
The number of binary operations on the set $\{1, 2, 3\}$ is $..........?$
- ✓$3^9$
- B$9^3$
- C$27$
- D$3!$
✓
Answer
Correct option: A.
$3^9$
Let us denote this set by $S$, then $\ce{∣S∣} = 3.$
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 $32 = 9$ 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 $3^9$.
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 $32 = 9$ 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 $3^9$.
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