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,.......a_m] and Q = [b₁,b₂,b₃......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

Mij = 0      if  (ai,bj) ∉ R
       1     if   (ai,bj )∈ R

Example

1. Let     P = {1, 2, 3, 4}, Q = {a, b, c, d}  
2. and     R = {(1, a), (1, b), (1, c), (2, b), (2, c), (2, d)}.  

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

1. A = {1, 2, 3, 4}  
2. R = {(1, 2) (2, 2) (2, 4) (3, 2) (3, 4) (4, 1) (4, 3)}  

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

1. Let P = {1, 2, 3, 4}  
2.     Q = {a, b, c, d}  
3.     R = {(1, a), (2, a), (3, a), (1, b), (4, b), (4, c), (4, d)  

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

1. Let P = {1, 2, 3, 4}   
2.     Q = {x, y, z, k}  
3.     R = {(1, x), (1, y), (2, z), (3, z), (4, k)}.  

The tabular form of relation as shown in fig: