Symmetric Relations
2026-02-28 15:48 Diff
https://brightchamps.com/en-us/math/algebra/symmetric-relations

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Last updated on October 28, 2025

A symmetric relation is a binary relation in which, if element a relates to element b, then b also relates to a. This means that the relationship is mutual and order-independent. This article further discusses symmetric relations.

What is a Relation in Math?

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A relation is a subset of the Cartesian product of two sets, connecting elements through ordered pairs. In each ordered pair, the first element comes from the domain, and the second from the range.

In other words, it connects elements of one set to elements of another through ordered pairs (a, b), where 'a' is from the domain and 'b' is from the range.

  • A = {x, y}
  • B = {3, 4, 5}
  • A relation between A and B could be R = {(x, 5), (y, 3)}
     

What are Symmetric Relations?

A symmetric relation on a set X means that whenever an ordered pair (a, b) is in R, then the pair (b, a) must also be included in R. Every connection between two elements will be mutual; if a relates to b, then b relates to a.

Examples

  1. "Is equal to" is an example of symmetric relations, such as $1 = 100 cents, and 100 cents = $1.
     
  2. "Is friend of " is also a symmetric relation.

Difference Between Asymmetric, Anti-symmetric, and Symmetric Relations

In mathematics, relations define connections between elements of sets using ordered pairs.

  • A relation is said to be symmetric if, whenever one element is related to another, the reverse is also true.
     
  • A relation is asymmetric if, whenever a is related to b, b is never related to a.
    Additionally, elements cannot relate to themselves.
  • A relation is antisymmetric if (a, b) ∈ R and (b, a) ∈ R imply a = b.

Here are some important points to remember. 

1. Asymmetric ⇒ Antisymmetric (and implies irreflexive).
 If xRy, then yRx cannot hold, so antisymmetry is satisfied.

2. Antisymmetric relations allow (a, a) ∈ R (e.g., ≤), but asymmetric relations do not.


3. Symmetric and asymmetric are mutually exclusive, except for the empty relation.


4. Symmetric and antisymmetric relations coincide only in trivial cases, such as the identity relation {(x, x) | x ∈ X}.

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Properties of Symmetric Relations

A relation R on a set A is symmetric if:
\(∀a, b∈A,  (a, b)∈R  ⟹  (b, a)∈R.\)

Equivalently, R = R-1(its inverse)


1. Inverse equals itself:
 R-1 = R.


2. Closed under set operations:
If R1 and R2 ​are symmetric, then so are R1 ∪ R2 and R1 ∩ R2​.


3. Matrix representation is symmetric:
Its adjacency matrix satisfies M = MT


4. Digraphs have paired edges:
Every directed edge a→b is accompanied by b→a.


5. Extremes are symmetric:
The empty relation and the universal relation (all pairs) are both symmetric.
 

Compatible with antisymmetry only trivially:

A relation is both symmetric and antisymmetric only in trivial cases, such as the identity relation {(x, x) | x ∈ X}.

Number of Symmetric Relations Formula

Number of symmetric relations = \(2^ { \ n \ ({ n \ + \ 1 \over 2})}\)


Where

  • n is the number of elements in the set,
  • Let N denote the number of symmetric relations.

Let's practice this using an example.
 

Example:  Set A = {1,2,3}:

Explanation: Here n = 3,


so, number of symmetric relations = \(2^ { \ 3\ ({ 3 \ + \ 1 \over 2})} = 2 ^ { 3 \times 2} = 2 ^ 6 = 64\)
 

Practice Problem: Similarly, find the number of symmetric relation of set B = {3, 2, 5, 3} by yourself for practice.

How to check if a Relation is Symmetric?

To check if a relation R on a set A is symmetric, follow these steps:

  1. List all ordered pairs in R, such as (a, b).
     
  2. For each pair, verify that the reverse (b, a) is also in R.
     
  3. If every pair has its reverse, the relation is symmetric.
     
  4. If any pair (a, b) ∈ R lacks its reverse (b, a) ∈ R, the relation is not symmetric.

Let's practice this using some problems.

Problem 1:
Let R = { (1, 2), (2, 1), (3, 4), (4, 3) }.
Each pair has its reverse, so R is symmetric.

Problem 2:
Let R = { (1, 3), (3, 4), (4, 3) }.
The pair (1, 3) does not have its reverse, so R is not symmetric.

Tips and Tricks to Master Symmetric Relations

To make symmetric relations easy and simple, here are a few quick tips and tricks:

  1. To check the symmetry, just flip the pair, and check if it exists in the relations.
     
  2. To find the transpose of a relation matrix, write the rows and columns and columns as rows.
     
  3. Use real life analogy for better understanding.
     
  4. Memorize the formula for calculating the number of symmetric relations, which is \(2^ { \ n \ ({ n \ + \ 1 \over 2})}\)
     
  5. Remember, every equivalence relation is symmetric, however the vice versa is not true.
     

Parent Tip: Relate symmetric relations to real life relations to help your child understand better. Encourage your child to practice. 

Common Mistakes of Symmetric Relations and How to Avoid Them

Students make mistakes while solving symmetric relations. Common mistakes like misidentifying asymmetric relations as symmetric and overlooking reversed pairs in data structures. Let’s check out some of these mistakes so that we can avoid them.
 

Real-Life Applications of Symmetric Relations

Symmetrical relations are prevalent in various aspects of our daily lives, from social interactions to mathematical concepts. Let’s see some of their real-life applications.

  1. Nature: Both butterfly wings have the same shades, designs and color; if one side has any pattern, then the other side will have the same. It is an example of symmetric relations.
  2. Biology: The relation ‘has similar DNA’ between two organisms is symmetric.
    For example, if a person has two children, the DNA's of both child are similar.
  3. Social Networks: In Neuroscience, the brain structure shows that the symmetry in both hemispheres deals with similar types of work; if one hemisphere is damaged, the other will also be affected.  
  4. Technology: Game design and graphics of computer simulation are used in mesh modelling, and character design for reduced and balanced compilation. 
  5. Art & Design: The software used in parametric design ensures symmetrical balance in 3D forms while building furniture, chairs, bench, or any architectures. 
    For example, if you are drawing a chair, you need to draw all legs of chairs of the same height.
     

Problem 1

Is the relation R = {(1, 1), (2, 2), (1, 2), (2, 1)} symmetric?

Okay, lets begin

Yes, the relation is symmetric.
 

Explanation

Checking the condition for symmetry

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

For each pair (a, b) ∈ R, the pair (b, a) is also in R. Since, it satisfies the condition for symmetry.

Thus, it is symmetric.

Well explained 👍

Problem 2

Is the relation R = {(1,1), (2, 2), (3, 3), (1, 2)} symmetric?

Okay, lets begin

No, the relation is not symmetric.
 

Explanation

Checking the condition for symmetry.

 R {(1,1), (2, 2), (3, 3), (1, 2)}

For symmetry, if (1,2) ∈ R, then (2,1) must also be in R. Since (2,1) ∉ R, the relation is not symmetric.
 

Well explained 👍

Problem 3

Is the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)} symmetric?

Okay, lets begin

No, the relation is not symmetric.
 

Explanation

The given relation is R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}

The pair (2, 3) ∈ R but (3, 2) ∉ R, violating the symmetry condition.

Thus, the given relation is not symmetric.

Well explained 👍

Problem 4

Is the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)} symmetric?

Okay, lets begin

Yes, the relation is symmetric.
 

Explanation

Checking the condition for symmetric

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

For every pair (a, b) ∈ R, the pair (b, a) is also in R, confirming symmetry.

Well explained 👍

Problem 5

Is the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (3, 1)} symmetric?

Okay, lets begin

No, the relation is not symmetric.

Explanation

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

The pair (3,1) ∈ R but (1,3) ∉ R, so the relation is not symmetric.

Well explained 👍

FAQs on Symmetric Relations

1.How to explain symmetric relation to my child?

A symmetric relation on a set ensures that if a is related to b, then b is related to a. 
 

Use examples like, you are related to your child, similarly your child is related to you. This relation is symmetric.
 

2.How can I help my child in learning symmetric relations?

Ask your child questions on symmetric relations by connecting it to real life. Like, ask if your child is younger than you, are you younger than your child as well, if you are married to your partner, is your partner married to you, if person A is taller than B, is B is taller than A, etc.

3.What example can I use to explain that "is less than" relation is not symmetric to my child?

Use parameters like height, weight, or age to explain. For example, if your child's weight is less than your weight, then your weight is less than your child. This is not true, hence it "is less than" is not a symmetric relation
 

4.How to explain to my child that a relation can be symmetric but not reflexive?

Use example like the relation "is married to” is symmetric because if a is married to b, then b is married to a. It is not reflexive because a person is not married to themselves.
 

5.What examples can I give to my child for of a symmetric relation?

  1. An example of a symmetric relation is “is a sibling of". If person A is a sibling of person B, then person B is also a sibling of person A.
     
  2. "is friend of" is also a symmetric relation. If Oliver is a friend of John, then John is also a friend of Oliver.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.