Symmetric Property
2026-02-28 23:39 Diff
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Last updated on October 23, 2025

The symmetric property is an algebraic concept that states, for any defined relation, if one element is related to another, then the second element is also related to the first. In this article, we will explore the symmetric property as it applies to equality, congruence, relations, and matrices.

What is Symmetric Property?

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In algebra, the symmetric property expresses that if one element in a set is related to another, then the second element is also related to the first in the same manner. This concept can be seen in several forms in mathematics, including:
Symmetric property of equality states that if x = y, then y = x

Symmetric property of congruence states: If in a set, one geometric figure is congruent to another, then the other figure is also congruent to the first one.

Symmetric property of relations says:  If a relation R is symmetric, then aRb ⇔ bRa, for all a, b.

Symmetric properties of matrices suggest:  If a matrix A is symmetric, then it is equal to its transpose A = AT
 

What is Relation in Math?

In mathematics, a relation shows how two or more values in a set are associated with each other. In an ordered pair with related elements, the first element is the domain and the second is the range. 

For example, consider the following sets:
P = {a, b}
Q = {1, 2, 3}

A relation R from P to Q could be:
R = (a, 2) (b, 1)
This indicates that a is related to 2, and b is related to 1.
This relation is a subset of the Cartesian product of two sets, P × Q.

How to check if a Relation Is Symmetric?

For a relation to be symmetric, each ordered pair in the given relation must satisfy the given condition:
(a, b) R (b, a) R

For example: 

Question: Consider a set A = 1, 2, 3 and a relation R on A defined as:
R = (1, 2), (2, 1), (2, 3), (3,2)
Now, let’s check if R is symmetric.
Solution: 
(1, 2) is in R → Is (2, 1) in R?  Yes


(2, 1) is in R → Is (1, 2) in R?   Yes


(2, 3) is in R → Is (3, 2) in R?   Yes


(3, 2) is in R → Is (2, 3) in R?  Yes

Since all pairs have a reverse pair in the relation, R is symmetric.
 

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Difference between Asymmetric and Symmetric Relations

Relations in mathematics are used as a way to show the connection between the elements of two sets. Among these relations are asymmetric and symmetric relations that differ from each other in the following ways:

Symmetric Relation

Asymmetric relation

Symmetric relations go both ways, meaning that two elements are related to each other in the same way.

A relationship is asymmetric when one element is related to another, but the other can never be related to the first one. The relationship goes only one way.

If (a, b) ∈ R, then (b, a)R

If (a, b)R, then (b, a)R (for all a  b)

The mathematical condition for symmetric relation is:
∀ a, b ∈ A, if (a, b) ∈ R ⇒ (b, a) ∈ R

The mathematical condition for asymmetric relation is:
∀ a, b ∈ A, if (a, b) ∈ R ⇒ (b, a) ∉ R

Symmetric relations can have self-pairs like (a, a)

Asymmetric relations do not contain self-pairs.

Example: R = (1, 2), (2, 1)

Example: R = (1, 2), (3, 4) but (2, 1) and (4, 3) are not present.

A real-life analogy of symmetric relations is: If A is the sibling of B, and B is a sibling of A

The real-life analogy of asymmetric relations is: If A is the parent of B, then B cannot be the parent of A.

What are the Properties of Symmetric Relations

The formula for a total number of symmetric relations with n elements is:

N = 2[n(n+1)]/2

Where, 
N is the number of symmetric relations, and n is the number of elements in the set.
Explanation:
The total number of possible ordered pairs from set A is: n2 =n × n

In a symmetric relation, if the pair (a, b) is included, then (b, a) must also be included. So, instead of selecting pairs freely, we select:

  • n diagonal elements (i.e., pairs like (a, a)): each can be included or not → 2 options per element.
     
  • n(n – 1)/2 off-diagonal element pairs, like (a, b) and (b, a): these come as pairs, and each pair either appears together or not at all → again 2 options per such pair.

Therefore, the total number of symmetric relations is:
2n × 2[n(n-1)]/2 = 2[n(n+1)]/2

For example: For a set A = 1, 2, n = 2

N = 2[n(n+1)]/2 = 23 = 8

So, for a 2-element set, there are 8 symmetric relations.
 

Common mistakes and How to Avoid Them in Symmetric Property

While applying the symmetric property in mathematics, students come across common misunderstandings working with ordered pairs or sets etc. Some of these are listed below, along with solutions to help avoid them: 
 

Real-life applications of Symmetric Property

As one learns about the symmetric property, it is also essential to understand its applications in real-world scenarios. Some of these are listed below.

  • Legal and business agreements: The symmetric property establishes that if a contract states that A owes B ‘x’ amount of money, then B is owed ‘x’ amount from A. This symmetric understanding ensures clarity in responsibilities and rights in legal or business transactions.
  • Currency Conversion: The symmetric property helps in converting currencies both ways. For instance, if $1 = ₹83, then ₹83 = $1.
  • Mirror Design in Architecture and Fashion:  Designers use symmetric properties to create balanced patterns in architecture and fashion. This means if one side (left) mirrors the other (right), making the design look the same from both directions.
  • Explaining Prescriptions to Patients: Doctors prescribe medicines in dosages. Using the symmetric property, they can explain the amount to their patients. For example, 5 mL = a teaspoon, so one teaspoon of medicine will be equal to 5 mL.
  • Assigning variables in coding : In coding, the symmetric property is used for comparing variables in search algorithms, data matching or conditional logic. 

Problem 1

If x = 12, use the symmetric property to rewrite this equation.

Okay, lets begin

 12 = x
 

Explanation

The equation remains true when the sides are swapped because of the symmetric property.
 

Well explained 👍

Problem 2

Given ∠P = ∠Q, what can you conclude using the symmetric property?

Okay, lets begin

 ∠Q = ∠P
 

Explanation

Symmetric property in geometric proofs suggests relationships between angles.
 

Well explained 👍

Problem 3

If 5x+3=y, use the symmetric property to rewrite this equation.

Okay, lets begin

y=5x+3
 

Explanation

 By the symmetric property of equality, if A = B, then B =  A.
So the equation can be written as y=5x+3

Well explained 👍

Problem 4

If triangle ABC is congruent to triangle DEF, what conclusion can we draw using the symmetric property?

Okay, lets begin

Triangle DEF is congruent to triangle ABC.
 

Explanation

In geometry, congruence is symmetric, so the order of triangles can be reversed.
 

Well explained 👍

Problem 5

If person A is listed as friends with person B in a database, what does the symmetric property suggest?

Okay, lets begin

 B is also friends with A.
 

Explanation

Friendship is a symmetric relation and goes both ways.

Well explained 👍

FAQs on Symmetric Property

1.Is a relation symmetric if it only contains self-pairs?

Yes, a relation containing only self-pairs is symmetric because each pair is its reverse.
 

2. Is symmetric property applicable for inequalities?

 No, the symmetric property is not applicable for inequalities. It is only applicable for equalities and symmetric relations. For example, the reverse of a > b cannot be b > a but a = b can be b = a in some cases.
 

3.Is the symmetric property only for numbers?

No, the symmetric property applies to any set where a relation is defined. For example, it can be applied to geometric figures (congruent triangles), words (is a synonym of) etc.
 

4.How is the symmetric property different from the transitive property?

The symmetric property is different from the transitive property in the following ways:

Symmetric Property

Transitive Property

The symmetric property is defined as: if element a is related to element b, then b is related to a.

The transitive property is defined as: if a is related to b, and b is related to c, then a is related to c.

Mathematically, If (a, b) ∈ R → then (b, a) ∈ R

Mathematically, If (a, b) ∈ R and (b, c) ∈ R → then (a, c) ∈ R

The direction of relation is both ways between the two elements.

The direction flows in a chain across three elements.

A minimum of one pair is required (a, b).

A minimum of two pairs are required ( a, b) (b, c)

On a graph, Arrows go in both directions between nodes

Visually, Arrows form a chain path leading from a to c.

5. Can the symmetric property be applied to inequalities?

 Most inequalities are not symmetric. If a < b, then b < a is not possible. So, the symmetric property does not apply to inequalities.

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.