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

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 $S.$