Question

The number of binary operations on $\left\{ 1,2,3,4 \right\}$ is _______________.
A. \[~{{4}^{2}}\]
B. \[~{{4}^{8}}\]
C. \[~{{4}^{3}}\]
D. \[~{{4}^{16}}\]

Answer 1

Hint: You've got the definition: a binary operation on S is any mapping from set to S. If you were to list
Then, you would need to find all possible ways to assign values to each possible pair \[\left( x,\text{ }y \right)\] where x and y are elements of S.
So, first, how many such pairs are there? There are \[2\] ways to pick the first, and \[2\] ways to pick the second, for a total of \[2\times 2=4\] pairs. They are, in fact \[\left( \text{a, a} \right)\text{, }\left( \text{a, b} \right)\text{, }\left( \text{b, a} \right)\text{, }\left( \text{b, b} \right)\text{.}\]
 Second, in how many different ways could you assign either a or b as the value of each of those? That will be the number of possible binary functions. We have \[2\] ways to assign a value to each of the \[4\] pairs; so
There are \[2\times 2\times 2\times 2=16\] ways.
So, the number of binary operations in a set with n elements $={{n}^{{{n}^{2}}}}$

Complete step by step solution:
Let ‘S’ be a finite set containing n elements.
In \[\left\{ 1,\text{ }2,\text{ }3,\text{ }4 \right\}\] there are \[4\] elements.
So, the number of binary operations in a set with n elements $={{n}^{{{n}^{2}}}}$
Here \[n=4\]
So the number of binary operations in a set with n elements \[={{4}^{({{4}^{2}})}}={{4}^{16}}\]
 So, binary operation is also a function from a set S is \[{{4}^{16}}\]

So, the correct answer is “Option D”.

Note: Even though one could define any number of binary operations upon a given nonempty set, we are generally only interested in operations that satisfy additional "arithmetic-like'' conditions. In other words, the most interesting binary operations are those that, in some sense, abstract the salient properties of common binary operations like addition and multiplication on \[S.\]