Table of Content

• What is Relation in Maths?
• What is Asymmetric Relations?
• Properties of Asymmetric Relations
• Asymmetric and Symmetric Relations
• Examples of Asymmetric Relations
• Conclusion: Asymmetric Relation
• Sample Problems on Asymmetric Relations
• FAQs On Asymmetric Relation

Asymmetric Relation Vs Symmetric Relations
Property	Asymmetric Relations	Symmetric Relations
Definition	(a, b) in the relation implies (b, a) is not in the relation	(a, b) in the relation implies (b, a) is also in the relation
Direction of Relationship	One-way relationship	Two-way relationship
Example	“Is Less Than” (<<)	“Is Equal To” (==), “Is a Friend of”
Transitivity	Can be transitive or intransitive	Often transitive
Matrix Representation	Zeros on the main diagonal, no symmetric entries	Symmetric entries reflected across the main diagonal
Antisymmetry	Always antisymmetric. (Not the same as asymmetric)	May or may not be antisymmetric
Irreflexivity	Always irreflexive	May or may not be irreflexive

Symmetric, Asymmetric and Antisymmetric

Symmetric: Symmetric Relation is a relation in which all x, y ∈ X, (x, y) ∈ R ⇒ (y, x) ∈ R

Asymmetric: Asymmetric Relation is a relation in which if for all x, y ∈ X, (x, y) ∈ R ⇒ (y, x) ∉ R

Antisymmetric: Antisymmetric Relation is a relation in which

• For all x, y ∈ X [(x, y) ∈ R and (y, x) ∈ R] ⇒ x = y
• For all x, y ∈ X [(x, y) ∈ R and x ≠ y] ⇒ (y, x) ∉ R

Examples of Asymmetric Relations

Example: Consider set A = {a, b}

• R = {(a, b), (b, a)} is not asymmetric relation but
• R = {(a, b)} is symmetric relation.

Example: Divisibility Relation:

• Consider the set of positive integers, and define the relation R as “divides.” If a divides b, then b cannot divide a.

R = {(2, 4), (3, 6), (5, 10)}

In this relation, each number on the left divides the corresponding number on the right, but the reverse is not true.

Example: Strict Subset Relation:

• Let’s define a set A and the relation R as “is a strict subset of.” If A is a strict subset of B, then B cannot be a strict subset of A.

R = {{1, 2}, {1, 2, 3}), {a, b}, {a, b, c}}

In this relation, each set on the left is a strict subset of the set on the right, but not vice versa.

Example:

• “Is Less Than” Relation:

Consider the relation “is less than” on the set of real numbers. If a<b, then it is not true that b<a. This is an asymmetric relation.

Example:

• “Is a Proper Subset Of” Relation:

Let’s consider the relation “is a proper subset of” on the set of all sets. If set A is a proper subset of set B, then it is not the case that B is a proper subset of A.

Example:

“Is Predecessor Of” Relation:

• Consider a relation representing the “is predecessor of” relation on the set of integers. If a is the predecessor of b, then it is not true that b is the predecessor of a, except when a=b+1.

Conclusion: Asymmetric Relation

In conclusion, an asymmetric relation is a specific type of binary relation on a set in which the order of elements matters. The key characteristic of an asymmetric relation is that if a pair (a,b) is in the relation, then the pair (b,a) must not be in the relation for any elements a and b from the set. In other words, the relationship is one-directional, ensuring that it does not allow for symmetry.

Read More,

• Set Theory
• Types Of Sets
• Relation and Function
• Types of Functions

Asymmetric Relation Example

Example: If A = {5, 9} is a relation R (a, b) on set A = (5, 9) ∈ R such that a < b, then prove that the relation is asymmetric.

Solution:

Given,

A = {5, 9} ∈ R

Here,

5 < 9, So A ∈ R

Now, for

(9, 5) 9 </ 5,

(5, 9) ∈ R ⇒ (9, 5) ∉ R

Thus, it is proved that the relation on set A is asymmetric

Sample Problems on Asymmetric Relations

Various problems on Asymmetric Relations are,

P1: Given a set {a, b, c}, determine whether the relation R = {(a, b), (b, c)} is asymmetric.

P2: Prove or disprove the statement: “The composition of two asymmetric relations is always asymmetric.”

P3: Let R₁ = {(1, 2), (2, 3)} and R₂ = {(3, 4), (4, 5)}. Find the composition R₁ ∘ R₂ and determine if it is an asymmetric relation.

P4: Provide an example of a relation that is irreflexive but not asymmetric.

FAQs On Asymmetric Relation

What is meant by asymmetric relation?

An asymmetric relation is a specific type of binary relation on a set where the order of elements matters. In an asymmetric relation, if the pair (a, b) is in the relation, then the pair (b, a) must not be in the relation for any elements a and b from the set. In other words, the relationship is one-directional or asymmetric.

What is an example of an asymmetric relationship?

An example of an asymmetric relationship can be found in the “parent” relationship

Is identity relation asymmetric?

No, the identity relation is not asymmetric. The identity relation on a set is defined in such a way that every element is related to itself.

Is An Empty relation asymmetric?

Yes, the empty relation is considered asymmetric. An empty relation on a set is one that contains no ordered pairs