What is Relation in Sets? Definition, Types and Representation | Class 10 Mathematics
Learn the definition of relations in sets, examples, types of relations, inverse relation, identity relation and different methods of representation with solved examples.
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| Relation in Sets – Definition, Types and Representation |
Introduction
Relations are one of the fundamental concepts of set theory and mathematics. A relation establishes a connection between the elements of two sets according to a specified rule or condition. Relations are widely used in mathematics, computer science, statistics, database systems, and many real-life applications.
For example, we may relate two numbers using conditions such as "is less than", "is greater than", "is divisible by", or "is equal to". Every such condition creates a mathematical relation between two sets.
What is a Relation?
Suppose A and B are two non-empty sets. If some elements of set A are associated with some elements of set B according to a particular rule, then the collection of all such ordered pairs is called a relation from A to B.
R={(x,y):x∈A, y∈B and xRy}
Here, xRy means that the element x is related to the element y according to the given condition.
Definition of Relation
A relation from a non-empty set A to another non-empty set B is any subset of the Cartesian product A×B.
R ⊆ A×B
Every ordered pair contained in the relation satisfies the condition used to define that relation.
Example of Relation
Let
A={3,4,5,8}
B={1,2,3,5,7}
Define the relation xRy whenever x<y.
Now check each element of set A with every element of set B. The ordered pairs satisfying the condition are
R={(3,5),(3,7),(4,5),(4,7),(5,7)}
This collection of ordered pairs represents the required relation from A to B.
Representation of Relations
A relation can be represented in several mathematical forms depending on the purpose of the problem. The most commonly used methods are:
- Rule or Set Builder Form
- Roster (Tabular) Form
- Graph on the Cartesian Plane
- Arrow Diagram
- Matrix Representation
1. Rule (Set Builder) Method
In this method, the relation is described by writing the rule that every ordered pair satisfies.
R={(x,y):x<y, x,y∈S}
Here every ordered pair belongs to the relation only if the first element is less than the second element.
2. Roster (Tabular) Method
In the roster method, all ordered pairs satisfying the given condition are listed explicitly.
Let
S={1,2,3,4,5}
If the relation is defined by x<y, then
R={(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)}
3. Graphical Representation
A relation can also be represented by plotting every ordered pair as a point on the Cartesian coordinate plane. Each plotted point represents one ordered pair belonging to the relation.
Example: For the relation R={(1,2),(1,3),(2,3)}, the points (1,2), (1,3) and (2,3) are plotted on the graph.
4. Arrow Diagram
An arrow diagram represents a relation by joining elements of one set with the corresponding elements of another set using arrows.
Each arrow indicates that the corresponding ordered pair belongs to the relation.
Types of Relations
Relations can be classified into different categories according to their properties. The most important types are:
- Inverse Relation
- Identity Relation
- Universal Relation
- Empty Relation
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
- Equivalence Relation
Inverse Relation
Let R be a relation from A to B. The inverse relation is obtained by interchanging the first and second elements of every ordered pair.
R-1={(b,a):(a,b)∈R}
Example
Let
A={l,m,n}
B={p,q,r}
R={(l,q),(l,r),(m,p),(n,p),(n,r)}
Then the inverse relation is
R-1={(q,l),(r,l),(p,m),(p,n),(r,n)}
Hence,
Domain of R = Range of R-1
Range of R = Domain of R-1
Identity Relation
A relation on a set A is called an Identity Relation if every element of the set is related only to itself.
I={(a,a):a∈A}
Example
If
A={a,b,c,d}
then
I={(a,a),(b,b),(c,c),(d,d)}
For an identity relation,
Domain = Range = A
Matrix Representation of Identity Relation
The identity relation can also be represented in the form of a matrix. In an identity matrix, every element on the principal diagonal is 1, while all other entries are 0.
| a | b | c | d | |
|---|---|---|---|---|
| a | 1 | 0 | 0 | 0 |
| b | 0 | 1 | 0 | 0 |
| c | 0 | 0 | 1 | 0 |
| d | 0 | 0 | 0 | 1 |
The diagonal entries are always 1, showing that each element is related only to itself, while every off-diagonal entry is 0.
Important Concepts
1. Relation
A relation from a set A to a set B is any subset of the Cartesian product A×B.
2. Ordered Pair
An ordered pair is written as (a,b), where the order of the elements is important.
3. Cartesian Product
The Cartesian product A×B is the set of all possible ordered pairs whose first element belongs to set A and the second element belongs to set B.
4. Domain
The domain of a relation is the set of all first elements of the ordered pairs.
5. Range
The range of a relation is the set of all second elements of the ordered pairs.
6. Inverse Relation
The inverse relation is obtained by interchanging the first and second elements of every ordered pair.
7. Identity Relation
An identity relation relates every element of a set only to itself.
8. Matrix Representation
A relation can be represented by a matrix in which 1 indicates the presence of an ordered pair and 0 indicates its absence.
Quick Revision
- ✔ A relation is a subset of A×B.
- ✔ Every relation consists of ordered pairs.
- ✔ A relation can be represented using Set Builder, Roster, Arrow Diagram, Graph and Matrix.
- ✔ The inverse relation is obtained by reversing every ordered pair.
- ✔ Identity relation contains only pairs of the form (a,a).
- ✔ Domain consists of first elements.
- ✔ Range consists of second elements.
- ✔ In an identity matrix, diagonal entries are 1 and all other entries are 0.
Exam Tips
- Always write the Cartesian product before constructing a relation.
- List only those ordered pairs that satisfy the given condition.
- Remember that the order of elements in an ordered pair cannot be changed.
- Do not confuse the domain with the range.
- For an identity relation, every ordered pair has identical first and second elements.
- The inverse relation is obtained simply by reversing every ordered pair.
- Practice drawing arrow diagrams and matrix representations for better understanding.
Frequently Asked Questions (FAQs)
1. What is a relation in mathematics?
A relation is a subset of the Cartesian product that connects elements of two sets according to a given rule.
2. What is the formula for a relation?
If R is a relation from A to B, then R ⊆ A×B.
3. What is the difference between a relation and a function?
A relation may associate one element with many elements, whereas a function assigns exactly one output to each input.
4. What is an inverse relation?
It is the relation obtained by reversing every ordered pair of the original relation.
5. What is an identity relation?
It is a relation in which every element of a set is related only to itself.
6. How can a relation be represented?
Relations can be represented using Set Builder Form, Roster Form, Arrow Diagram, Graph and Matrix Representation.
7. What is the domain of a relation?
The domain is the set of all first elements of the ordered pairs.
8. What is the range of a relation?
The range is the set of all second elements of the ordered pairs.
Conclusion
Relations are one of the most important concepts in set theory and form the foundation of advanced topics such as functions, mappings, graphs, and database systems. Understanding the definition of relations, their representations, and different types such as inverse and identity relations makes solving mathematical problems much easier. By practicing examples and learning the various methods of representation, students can build a strong conceptual understanding for board examinations and competitive exams.
References
- NCERT Mathematics Textbook – Class 10
- ASSEB Mathematics Textbook
- Relations and Functions Chapter
- CBSE Class 10 Mathematics Curriculum
- Standard Higher Secondary Mathematics Reference Books