Question

Let $A = \\{ a,b\\}$ . List all relations on $A$ and find their number.

Answer 1

Here we need to find the relation of the given set. We will first write the given set as ordered pairs. We will treat this list of possible ordered pairs as a set, and so each relation will consist of a subset of them. Then we will find the number of subsets and hence the total possible number of relations using this.

Complete step-by-step answer:
Here we need to find the relation on $A$.
Here, $A = \\{ a,b\\}$
Any relation on $A$ can be written as a set of ordered pairs.
The ordered pairs which is denoted by $A \times A$ can possibly be included are
A \times A = \left\\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {b,a} \right),\left( {b,b} \right)} \right\\}
We can see that the number of elements in set $A \times A$ is 4.
We know the formula number of subsets of any set is equal to ${2^n}$, here $n$ is the number of elements present in the set.
Now, we will find the number of subsets of set $A \times A$ . We know that the number of elements in $A \times A$ is 4.
Therefore, number of subsets of set $A \times A$ is equal to ${2^4}$
On applying the exponent on the base, we get
$\Rightarrow$ Number of subsets of set $A \times A$ $= 2 \times 2 \times 2 \times 2 = 16$
Therefore, number of relations on set $A$ is 16.
Now, we will list all possible relations on set $A$. Therefore, we get
\begin{array}{l}\left\\{ {} \right\\},\left\\{ {\left( {a,a} \right)} \right\\},\left\\{ {\left( {a,b} \right)} \right\\},\left\\{ {\left( {b,a} \right)} \right\\},\left\\{ {\left( {b,b} \right)} \right\\},\\\\\left\\{ {\left( {a,a} \right),\left( {a,b} \right)} \right\\},\left\\{ {\left( {a,a} \right),\left( {b,a} \right)} \right\\},\left\\{ {\left( {a,a} \right),\left( {b,b} \right)} \right\\}\\\\\left\\{ {\left( {a,b} \right),\left( {b,a} \right)} \right\\},\left\\{ {\left( {a,b} \right),\left( {b,b} \right)} \right\\},\left\\{ {\left( {a,a} \right),\left( {a,b} \right)} \right\\},\\\\\left\\{ {\left( {b,a} \right),\left( {b,b} \right)} \right\\},\left\\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {b,a} \right)} \right\\},\\\\\left\\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {b,b} \right)} \right\\},\left\\{ {\left( {a,a} \right),\left( {b,a} \right),\left( {b,b} \right)} \right\\},\\\\\left\\{ {\left( {a,b} \right),\left( {b,a} \right),\left( {b,b} \right)} \right\\},\left\\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {b,a} \right),\left( {b,b} \right)} \right\\}\end{array}

Note: Here we obtained the relation on the set $A$. In set theory, a relation between two sets is a subset of their Cartesian product and function is a special type of relation. We can say that a function is always a relation, but a relation can’t always be a function. The term set is defined as the collection of numbers which are well defined and which either follow a specific rule or are selected randomly.