### Theory:

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

If you're answer is

**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**${2}^{3\times 2}={2}^{6}$.

**Now we will practice some exercises based on the concepts we discussed**.