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Table of contents
1.
Introduction
2.
5 Representations of Relations
2.1.
Ordered Pair
2.2.
Relation as a Matrix
2.3.
Mapping Diagram
2.4.
Digraph
2.5.
Relation as a Table
3.
FAQs
4.
Key Takeaways
Last Updated: Mar 27, 2024
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Representation of Relations

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Introduction

Given any two non-empty sets A and B, A relation R from A to B is a subset of the Cartesian product A x B and is derived by describing a relationship between the first element (say x) and the other element (say y) of the ordered pairs in A & B. Elements corresponding to x are called the domain of the given ordered pairs. In contrast, elements corresponding to y are called the range of the given ordered pairs. 

 

5 Representations of Relations

Ordered Pair

In this set of ordered pairs of x and y are used to represent relation. In this, corresponding values of x and y are represented using parenthesis.

 

Example:

S: {(1,2), (2,4), (3,6)} Set of Ordered Pairs

 

Relation as a Matrix

In this, zero-one is used to represent the relationship between two sets. In this, if an element is present, then it is represented by 1; else, it is represented by 0. This method makes it easy to judge if a relation is reflexive, symmetric, or transitive just by looking at the matrix.

 

Suppose R is a relation from X={x1, x2, .....xn} to Y={y1, y2....yn}

It is represented by :-

 

M[i, j]={1, if (Xi, Yj) belongs to R

        0, if (Xi, Yj) does not belong to R}

 

If A={1, 2, 3} and B={1, 2} and Relation R is

R = {(1, 2) (2, 1), (3, 1), (3, 2)}

then all corresponding value of Relation will be represented by “1” else “0”.

It is represented as:

 

 

Mapping Diagram

A mapping or an arrow diagram is a visual representation of a relation.

If S and T are finite sets and R is a relation from S to T. Let's see how a Relation R can be depicted using a mapping diagram.

 

Draw two ellipses for the sets S, and T. Write down the elements of S and elements of T column-wise in ellipses. Then draw a directed arrow from the first ellipse to the second ellipse if s is related to t and s ∈ S and t ∈ T.

 

Let S = {1, 2, 3, 4}  

T = {s, t, u, v}  

R = {(1, s), (2, s), (3, s), (1, t), (4, t), (4, u), (4, v)}

 

 

Digraph

A relation can be represented using a directed graph(Digraph).

The number of vertices in the graph represents the number of elements in the set from which the relation has been defined. For each ordered pair, say (x, y) in the relation R, there will be a directed edge/arrow from the vertex 'x' to vertex 'y'. 

Also, if there is an ordered pair (x, x), there will be a self-loop on vertex 'x.'

 

Example:

A = {1, 2, 3, 4}  

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

 

 

Relation as a Table

A relation can be represented using a table. Relation R can be represented in a tabular form. It presents the elements of domain and range in a tabular way. 

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

 

Let S = {1, 2, 3, 4}   

T = {s,t,u,v}  

R = {(1, s), (1, t), (2, u), (3, u), (4, v)}.

 

 

 

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FAQs

1. What do you mean by a relation?

The relation is defined as the relationship between two or more sets of values. Let's suppose x and y are two sets of ordered pairs. And set x has relation with set y, then the values of set x are called domain, and the values of set y are called range.

 

2. What is the difference between a relation and a function?

The basic factor in differentiating between the relation and function is that a function has a single input for a single output, but a relation can have many outputs for a single input. 

 

3. How can a relation be represented?

A relation can be represented using ordered pairs, a matrix, digraph, mapping diagram, and table.

 

Key Takeaways

In this blog, we have extensively shown how to represent a relation differently, like using Diagraph, Matrix, and ordered pairs. Which do you think is more convenient to use among these different representations? Think wisely according to the data!!!.

 

For the practice problems related to the relations in Maths, you can visit here.

 

Visit Coding Ninjas Studio to prepare the best for the big tech interview.

 

Happy Learning!!!

 

 

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