Question

Find the total number operations in the set $S=\left\{ a,b,c \right\}$.\[\]

Answer 1

Hint: Find out the number of elements of the domain and co-domain set of the binary operation. Use the formula total number of operations that can be made between two sets to proceed. \[\]

Complete step by step answer:
 A binary operation is a map which combines two values and returns one value. It means it sends a Cartesian order pair to a single element. The set from where the operation takes inputs to combine is called domain and the set from where the operation returns outputs is called co-domain. \[\]
Let us define a map $f$such that $f$ is mapped from the domain set A to the co-domain set B. We assign the number of elements of A is $a$ and B is $b$ . Now according to the definition of a map every element of A is mapped and one element of A is not mapped more than element of B. So every element of B has $a$ choice in the set A to b mapped from and every element of A has $b$ choices to map. So total number of such maps is
\[b\times b\times ...\left( a\text{ times} \right)={{b}^{a}}\]
The given set in the question is $S=\left\{ a,b,c \right\}$ which has three elements. Let us denote any binary operation the set as $o$ such that
     \[\begin{align}
  & o:S\times S\to S \\
 & \Rightarrow o:\left\{ a,b,c \right\}\times \left\{ a,b,c \right\}\to S \\
\end{align}\]
 If we write the set $S\times S$ in the list form we are going to get $3\times 3=9$ ordered pairs. The elements of the set $S\times S$ is
$S\times S=\left\{ \left( a,a \right),\left( b,b \right),\left( c,c \right),\left( a,c \right),\left( c,a \right),\left( a,b \right),\left( b,a \right),\left( b,c \right),\left( c,b \right) \right\}$
The binary operation $o$ sends a pair from the set $S\times S$ to $S$. The set $S\times S$ is the domain set of the operation $o$ has 9 elements and $S$ is the co-domain set of $o$ which has 3 elements. We use the formula total number of maps between two sets. The elements from set $S\times S$ has 3 choices from set $S$.

So total number of operations is \[3\times 3\times ...\left( 9\text{ times} \right)={{3}^{9}}=19863\]

Note: We need to find properly the cardinality of the domain set and co-domain set. You can also use the formula of total number of binary operations on set with $n$ number of elements which is ${{n}^{{{n}^{2}}}}$. We can use it in this problem and find the total number of operations as ${{3}^{{{3}^{2}}}}={{3}^{9}}=19863$.