Symmetric Relations

Symmetric relation is a binary relation which satisfies that if aRb exists then bRa also exists for all a, b belongs to set S. 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.

In this article, we will explore Symmetric relations, Symmetric relation definition, properties of Symmetric relations, and Symmetric Relations Examples. We will also solve some problems related to Symmetric relations. Let’s start our learning on the topic ” Symmetric Relation”.

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 = {x, y}
• B = {3, 4, 5}

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.,

• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Irreflexive Relation
• Asymmetric Relation
• Antisymmetric Relation
• Equivalence Relation

What are Symmetric Relations?

A relation is called Symmetric relation when 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,

• R = {(7, 9), (9, 7)}

Definition of Symmetric Relations

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. M_R = (M_R)^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 for the number of symmetric relations with n-elements is given by:

Number of Symmetric Relation = 2^{[n(n -1)]/2}

Symmetric Relation Formula

For a set having ‘n’ number of elements number of symmetric relation is given as:

 N = 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: If a > b then b /> a	Example: a + b = b + a

Conclusion

From the above discussion we can conclude that a relation R is said to be a symmetric relation when if x and y holds the relation R, y and x also holds the relation R i.e., if xRy then yRx. The formula for calculating the total number of symmetric relations from a set of n elements is 2^{[n(n -1)] / 2}.

Read More:

• Relation
• Domain and Range of Relation

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 = 2²

Total number of symmetric relation on given set V = 4

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?

FAQs on Symmetric Relations

What is Symmetric Relation?

A relation is called a symmetric relation if a is related to b then, b is also related to a where, a and b are two elements of any set and are related with relation R.

What is an Example of Symmetric Relation?

An example of symmetric relation is addition of two numbers.

Is an Antisymmetric Relation Always Symmetric Relation?

No, an antisymmetric relation is not always a symmetric relation.

Is Null Set a Symmetric Relation?

Null set is a symmetric relation for every set.

What is the Formula for Number of Symmetric Relation?

Formula for number of symmetric relations is given by:

Number of Symmetric rRelation = 2^{[n(n -1)] / 2}