### Theory:

We acquired an elaborated idea of sets, domain, range and other fundamental concepts. Keeping all these in mind, try to answer this question.

Should all sets have a relation?

If you're answer is No, you're absolutely correct. Some sets do not have a relation, and those are named empty sets or null sets.
Null relation:
When there is no relation between any set elements, that relation is called a null set or an empty set.
Example:
If set $$A$$ denotes the states of India and $$B$$ denotes the capital of the Indian states.

That is $$A = \{Kerala, Gujarat, Punjab\}$$ and $$B = \{Chennai, Mumbai, Kolkata\}$$.

We can notice here, the set $$A$$ and $$B$$ (states and capital) contain no relation $$R$$.

Thus $$R$$ contain no element. Therefore, a relation which contains no element is called a “Null relation”.
Null relation is denoted by $\varnothing$. Here, $$R =$$ $\varnothing$. Similarly, an empty set is represented by $$\{\}$$. Here, $$R = \{\}$$
Total number of relations
Do you know how to find the total number of relations? Now we will learn how to calculate it.
If the number of elements in the set $$A$$ $$= n(A)$$ $$= m$$ , then the number of elements in the set $$B$$ is $$=n (B)$$ $$= n$$, then the total number of relations that exist from $$A$$ to $$B$$ is $$2^m$$$$^n$$.
Example:
Let the two sets $$A = \{(4, 8, 13\}$$ and $$B = \{(15, 11\}$$.

Here, $$n (A) = 3$$ and $$n (B) = 2$$.

Therefore, total number of relations that exist from $$A$$ to $$B$$ is ${2}^{3×2}={2}^{6}$.
Now we will practice some exercises based on the concepts we discussed.