# Definition of Relations

Let A and B be two nonempty sets. A relation R from A to B is a subset of A × B.

If R Í A × B and (a, b) ∈ R we say that a is related to b by R and we write a R b. If a is not related to b by R we write a b.

In case A and B are the same set, i.e. if R Í A × A, we say R is a relation on A or a relation from A to A.

**Illustration**

Let A = {1, 2, 3} and B = {3, 5, 7}. Write the relation R from A to B such that x R y if x < y.

**Solution**

Since 1 ∈ A is less than 3, 5, 7 ∈ B, (1, 3), (1, 5), (1, 7) ∈ R.

Similarly, (2, 3), (2, 5), (2, 7) ∈ R.

For 3 ∈ A, as 3 = 3, (3, 3) Ï R.

But since 3 < 5, 3 < 7, (3, 5), (3, 7) ∈ R.

Hence R = {(1, 3), (1, 5), (1, 7), (2, 3), (2, 5), (2, 7), (3, 5), (3, 7)}.

**Examples**

(i) Let A = {1, 2, 4, 5}. Let R be a relation on A defined by x R y if x divides y.

**Solution**

As 1 divides every natural number, 1

*R*x for each x ∈ A. That is, (1, 1), (1, 2), (1, 4), (1, 5) ∈ R.

Also, as 2 divides 2 and 2R2 and 2R4. Since 2 does not divide 1 and 5, (2, 1), (2, 5) Ï R.

Similarly (4, 4) and (5, 5) ∈ R.

Thus, R = {(1, 1), (1, 2), (1, 4), (1, 5), (2, 2), (2, 4), (4, 4), (5, 5)}.

**(ii) Let A = {1, 2, 4, 5, 8, 64, 625}. Define R on A by x**

*R*y if x^{3}= y.**Solution**

(ii) Since 1 = 1

^{3}, 2

^{3}= 8, 4

^{3}= 64 and 5

^{3}= 625,

R = {(1, 1), (2, 8), (4, 64), (5, 625)}.