**Question:**

Let A and B be two nonempty sets.

(i) What do you mean by a relation from A to B?

(ii) What do you mean by the domain and range of a relation?

**Solution:**

(i) If $A$ and $B$ are two nonempty sets, then any subset of the set $(A \times B)$ is said to a relation R from set A to set B.

That means, if $R$ be a relation from $A$ to $B$ then $R \subseteq(A \times B)$.

Therefore, $(x, y)^{\in} \mathrm{R} \Rightarrow(x, y)^{\in}(A \times B)$

That means x is in relation to y. Or we can write xRy.

(ii) Let R be a relation from A to B. Then, the set containing all the first elements of the ordered pairs belonging to R is called Domain.

For the relation $R, \operatorname{Dom}(R)=\{x:(x, y) \in R\}$

And the set containing all the second elements of the ordered pair belonging to R is called Range.

For the relation $R$, Range $(R)=\left\{y:(x, y)^{\in} R\right\}$