# Representation of Relations

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

1. Relation as a Matrix: Let P = [a1,a2,a3,.......am] and Q = [b1,b2,b3......bn] 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 = [Mij], defined as

```Mij = 0      if  (ai,bj) ∉ R
1     if   (ai,bj )∈ 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 ∈ P and b ∈ 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: