Asymmetric Relation

A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on. A relation is a subset of the cartesian product of a set with another set. A relation contains ordered pairs of elements of the set it is defined on.

Relation in Mathematics is defined as the relationship between two sets. If we are given two sets set A and set B and set A has a relation with set B then each value of set A is related to a value of set B through some unique relation. Here, set A is called the domain of the relation, and set B is called the range of the relation.

For example if we are given two sets, Set A = {1, 2, 3, 4} and Set B = {1, 4, 9, 16} then the ordered pair {(1, 1), (2, 4), (3, 9), (4, 16)} represents the relation defined as, R, A: → B {(x, y): y = x²: y ϵ B, x ϵ A}.

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.

Asymmetric Relations Definition

A relation R on a set A is called asymmetric relation if 

∀ a, b ∈ A, if (a, b) ∈ R then (b, a) ∉ R and vice versa,

where R is a subset of (A x A)

This if an ordered pair of elements “a” to “b” (_aR_b) is present in relation R then an ordered pair of elements “b” to “a” (_bR_a) should not be present in relation R.

Some of the properties of Asymmetric Relations are:

• Empty relation on any set is always asymmetric.
• Every asymmetric relation is also irreflexive and anti-symmetric.
• Universal relation over a non-empty set is never asymmetric.
• A non-empty relation can not be both symmetric and asymmetric.
• An asymmetric relation is always irreflexive, meaning no element is related to itself.
• Asymmetric relations can be represented by directed acyclic graphs (DAGs).
• Asymmetric relations are often used in the context of partial orderings, where the relation represents a partial order if it is antisymmetric and transitive.
• While asymmetric relations are not necessarily antisymmetric, antisymmetric relations are always asymmetric.

An asymmetric relation is a binary relation on a set where if (a,b) is in the relation, then (b,a) must not be in the relation for any elements a and b.

Example:

• The “is a parent of” relationship is asymmetric. If Alice is the parent of Bob, then Bob cannot be the parent of Alice.

A symmetric relation is a binary relation on a set where if (a, b) is in the relation, then (b,a) must also be in the relation for any elements a and b.

Example:

The “is a sibling of” relationship is symmetric. If Alice is a sibling of Bob, then Bob is also a sibling of Alice.

Difference Between Asymmetric and Symmetric Relations

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

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

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

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.

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