# Representation of Relations

Relations can be represented in many ways. Some of which are as follows:

**1. Relation as a Matrix:** Let P = [a_{1},a_{2},a_{3},â€¦â€¦.a_{m}] and Q = [b_{1},b_{2},b_{3}â€¦â€¦b_{n}] are finite sets, containing m and n number of elements respectively. R is a relation from P to Q. The relation R can be represented by m x n matrix M = [M_{ij}], defined as

M_{ij}= 0 if (a_{i},b_{j}) âˆ‰ R 1 if (a_{i},b_{j})âˆˆ R

**Example**

The matrix of relation R is shown as fig:

**2. Relation as a Directed Graph:** There is another way of picturing a relation R when R is a relation from a finite set to itself.

**Example**

**3. Relation as an Arrow Diagram:** If P and Q are finite sets and R is a relation from P to Q. Relation R can be represented as an arrow diagram as follows.

Draw two ellipses for the sets P and Q. Write down the elements of P and elements of Q column-wise in three ellipses. Then draw an arrow from the first ellipse to the second ellipse if a is related to b and a âˆˆ P and b âˆˆ Q.

**Example**

The arrow diagram of relation R is shown in fig:

**4. Relation as a Table:** If P and Q are finite sets and R is a relation from P to Q. Relation R can be represented in tabular form.

Make the table which contains rows equivalent to an element of P and columns equivalent to the element of Q. Then place a cross (X) in the boxes which represent relations of elements on set P to set Q.

**Example**

The tabular form of relation as shown in fig: