Symmetric Relations | Types definition and Examples
Last Updated : 28 Apr, 2025
Symmetric relation is a type binary relation in discrete Mathematics defined on sets. Which satisfies that if aRb exists then bRa also exists for all pairs of a, b that belong to set S. which means that if (a, b) belongs to R then (b, a) Also belongs to relation R.
Example of symmetric relation includes "is equal to", as if a = b is true then b = a is also true.
This article will explore Symmetric relations' definitions, properties, and Examples. Along with some solved and unsolved problems related to Symmetric relations.
What is Relation in Math?
Relation represents the association of two or more values in the set. If the two values of an ordered pair are related then, the first value in the ordered pair is called the Domain and the second value in the ordered pair is called the range. It is the subset of the cartesian product of two sets.
For example, let's consider two sets:
A relation between A and B could be R = {(x, 5), (y, 3)}
Types of Relation
There can be various types of relations in mathematics, i.e.,
What are Symmetric Relations?
Symmetric relation are a type of relations where the two elements of set X are related with relation R then reversing the order of the elements is also related with the relation R.
In other words, symmetric relation is defined as if xRy then yRx where x and y are two element of set S and R is relation. A relation R = {(x, y) → R | a + b} is a symmetric relation.
For example, A = {7, 9} then symmetric relation R on A if,
Symmetric Relation Definition
The relations are said to be symmetric if in a set S the two elements a and b, if a is related to b then, b is also related to a. Also, if for every (a, b) belongs to relation R then, (b, a) also belongs to relation R i.e., if (a, b) ∈ R then (b, a) ∈ R.
If p and q are two elements of set S related with relation R then, conditions for relation to be symmetric:
pRq ⇔ qRp ∀ p, q ∈ S
Examples of Symmetric Relations
There are multiple examples of symmetric relation. Some of these examples are listed below:
- Addition of two elements
- Multiplication of two elements
- Equality relation on any set.
Properties of Symmetric Relations
Some properties of symmetric relation are listed below:
- Empty relation on any set is always symmetric.
- Universal relation is always symmetric.
- If R is a symmetric relation, then R-1 is also symmetric.
- If R1 and R2 are symmetric relations, then R1 ∪ R2 is also symmetric.
- If R1 and R2 are symmetric relations, then R1 ∩ R2 is also symmetric.
- A relation can be symmetric and antisymmetric at same time.
- A relation cannot be symmetric and asymmetric at same time.
- In the matrix representation of the symmetric relation, the transpose of the matrix is equal to the original matrix. MR = (MR)T.
- In the directed graph representation of the symmetric relation, if there is an edge between two distinct nodes then, an opposites edge is also present between the two nodes.
Number of Symmetric Relations Formula
Formula for the total number of symmetric relations with n-elements is given by:
Number of Symmetric Relation = 2[n(n +1)]/2
where,
- N is Number of Symmetric Relations
- n is Number of Elements in Set
How to Check Relation is Symmetric or Not?
To check whether the given relation is symmetric or not follow the below steps.
- First check if (a, b) is present in the relation.
- If (a, b) is present and then check for (b, a).
- If (b, a) is present then, relation is symmetric.
- If (b, a) is absent then, relation is not symmetric.
Asymmetric and Symmetric Relations
Below table represents the difference between the symmetric and asymmetric relation.
Characteristics | Symmetric Relation | Asymmetric Relation |
|---|
Definition | A relation R is symmetric when two elements p and q of set A if p is related to q, then q is also related to p. | A relation R is asymmetric when two elements p and q of set A if p is related to q then, q is not related to p. |
|---|
Mathematical Representation | pRq ⇔ qRp or (p, q) ∈ R then, (q, p) ∈ R | pRq ⇔ q not related p (p, q) ∈ R then, (q, p) ∉ R |
|---|
Example | Addition if a + b then b + a. | Division if a / b the b ∤ a. |
|---|
Asymmetric, Anti-Symmetric and Symmetric Relations
Difference between the asymmetric, anti-symmetric and symmetric relations
Asymmetric Relations | Anti-Symmetric Relations | Symmetric Relations |
|---|
Relation R on a set A is said to be asymmetric if and only if (a, b) ∈ R, then (b, a) ∉ R, for all a, b ∈ A. | Relation R on a set A is said to be antisymmetric, if aRb and bRa hold if and only if when a = b. | Relation R is said to be symmetric iff, for elements a, b ∈ A, we have aRb, that is, (a, b) ∈ R, then we must have bRa, that is, (b, a) ∈ R. |
Example: a - b ≠ b - a | Example: | Example: a + b = b + a |
Conclusion
Symmetric relations play a crucial role in the study of discrete mathematics, offering a foundation for understanding more complex structures like equivalence relations and graph theory. By ensuring that every relationship is bidirectional, symmetric relations model a wide range of real-world scenarios, from social networks to symmetric matrices in linear algebra.
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Symmetric Relations Examples
Example 1: Check whether the relation R = {(2,5), (3,3)} is symmetric or not?
Solution:
R = {(2,5), (3,3)}
Above relation is not a symmetric relation as:
(2, 5) ∈ R but (5, 2) ∉ R
R is not symmetric.
Example 2: Prove that given relation R = {(1,2), (2,1), (4,4), (5,7), (7, 5)} is symmetric relation?
Solution:
R = {(1,2), (2,1), (4,4), (5,7), (7, 5)}
Above relation is symmetric relation as:
(1, 2) ∈ R then, (2, 1) ∈ R
(2, 1) ∈ R then, (1, 2) ∈ R
(4, 4) ∈ R then, (4, 4) ∈ R
(5, 7) ∈ R then, (7, 5) ∈ R
(7, 5) ∈ R then, (5, 7) ∈ R
R is symmetric.
Example 3: Find the number of symmetric relations in set V with 3 elements.
Solution:
Total number of symmetric relation = 2[n(n +1)] / 2
Total number of symmetric relation on given set V= 2[3(3 +1)] / 2
Total number of symmetric relation on given set V = 26
Total number of symmetric relation on given set V = 64
Practices Question on Symmetric Questions
Q1: Find the number of symmetric relations in set A with 9 elements.
Q2: Prove that given relation R = {(4, 5), (7, 8), (9 ,1), (1, 9), (8, 7)} is symmetric relation?
Q3: Check whether the relation R = {(2,5), (3,3)} is symmetric or not?