Question

Find the total number of binary operations on $\{a,b\} $.

Answer 1

Hint: Here we will proceed to the solution by knowing the formula for the number of binary operations possible for a given number of elements in the set.

Complete Step-by-Step solution:
Let us assume the given set as $S$ with two elements $a$ and $b$.
So the set can be represented as $S = \{a,b\} $
Here we have to find the total number of binary operations possible for the given set $S$.
We know that binary operation on a set is nothing but calculation that combines two numbers to produce another element.
We know that if there is any set $S$ with $n$ elements then the total number of binary operation is given with a formula i.e.
Number of binary operations =${\left( n \right)^{{n^2}}}$
Here $n$ is the number of elements in the given set
Now we know the given set $S$ has two elements that means n=2
 So, Number of binary operations for given set $S$=${\left( 2 \right)^{{2^2}}}$=16 $[\because {2^2} = 4{ \text{ and }}{2^4} = 16]$

NOTE: For the problems like above mentioned we should know what is the operation given and on which we have to perform this operation. So in the above problem binary operation is a simple calculation that combines two numbers to produce another number. But if we clearly observe we have to apply the operation on sets where it is important that the domain and codomain should be of the same set.