Symmetric relations are a type of relation 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, a symmetric relation is defined as if xRy then yRx, where x and y are two elements of set S and R is a relation. A relation R on a set A is symmetric if, whenever (x, y) ∈ R, then (y, x) ∈ R.
For example, A = {7, 9} then symmetric relation R on A if,
- R = {(7, 9), (9, 7)}
Examples of Symmetric Relations
There are multiple examples of a 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 a symmetric relation are listed below:
- An empty relation on any set is always symmetric.
- A 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 the same time.
- A relation cannot be symmetric and asymmetric at the 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 opposite edge is also present between the two nodes.
Number of Symmetric Relations Formula
The 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 the Number of Symmetric Relations
- n is Number of Elements in Set
How to Check Relation is Symmetric or Not?
- 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 vs Symmetric Relations
The below table represents the difference between the symmetric and asymmetric relations.
Symmetric Relation | Asymmetric Relation |
|---|---|
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. |
pRq ⇔ qRp or (p, q) ∈ R then, (q, p) ∈ R | pRq ⇔ q not related p (p, q) ∈ R then, (q, p) ∉ R |
Addition if a + b then b + a. | Division if a / b the b ∤ a. |
Asymmetric vs Anti-Symmetric vs Symmetric Relations
Difference between the asymmetric, antisymmetric and symmetric relation
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 if, 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 |
Also Check
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: Check whether the relation R = {(2, 5), (3, 3)} is symmetric or not?
Q2: Prove that given relation R = {(4, 5), (7, 8), (9 ,1), (1, 9), (8, 7)} is symmetric relation?
Q3: Find the number of symmetric relations in set A with 9 elements.