Asymmetric Relation

In discrete Maths, an asymmetric relation is just opposite to symmetric relation. In a set A, if one element less than the other, satisfies one relation, then the other element is not less than the first one. Hence, less than (<), greater than (>) and minus (-) are examples of asymmetric. We can also say, the ordered pair of set A satisfies the condition of asymmetric only if the reverse of the ordered pair does not satisfy the condition. This makes it different from symmetric relation, where even if the position of the ordered pair is reversed, the condition is satisfied. Apart from symmetric and asymmetric, there are a few more types of relations, such as:

  • Empty Relation
  • Universal Relation
  • Identity Relation
  • Inverse Relation
  • Reflexive Relation
  • Symmetric Relation
  • Transitive Relation
  • Equivalence Relation

Domain and Range: If there are two sets X and Y and Relation from X to Y is R(x,y), then domain is said to be as the set { x | (x,y) ∈ R for some y in Y} and Range is said to be as the set {y | (x,y) ∈ R for some x in X}.

Asymmetric Relation

In set theory, A relation R on a set A is called asymmetric if no (y,x) ∈ R when (x,y) ∈ R. Or we can say, the relation R on a set A is asymmetric if and only if, (x,y)∈R⟹(y,x)∉R. For example:

If R is a relation on set A = {12,6} then {12,6}∈R implies 12>6, but {6,12}∉R, since 6 is not greater than 12.

Note: Asymmetric is the opposite of symmetric but not equal to antisymmetric.

Example: The operators -, < and > are examples of asymmetric, whereas =, ≥, ≤, is TwinOf() does not satisfy the asymmetric condition.

Also, read:

Properties of Asymmetric Relation

  • A relation is said to be asymmetric if it is both antisymmetric and irreflexive or else it is not.
  • Limitations and opposites of asymmetric relations are also asymmetric relations. For example, the inverse of less than is also asymmetric.
  • A transitive relation is asymmetric if it is irreflexive or else it is not. Suppose if xRy and yRx, transitivity gives xRx, denying ir-reflexivity.
  • As a result, if and only if, a relation is a strict partial order, then it is transitive and asymmetric.
  • Not all asymmetric relations are strict partial orders.
  • An asymmetric relation must not have the connex property. For example, the strict subset relation ⊊ is asymmetric and neither of the sets {3,4} and {5,6} is a strict subset of the other.

Asymmetric Relation Example

Example: If A = {2,3} and relation R on set A is (2, 3) ∈ R, then prove that the relation is asymmetric.

Solution: Given A = {2,3} and (2, 3) ∈ R

Clearly, 2 is less than 3, 2<3, but 3 is not less than 2, hence,

(2, 3) ∈ R ⇒ (3,2) ∉ R

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

Comparison With Symmetric and Antisymmetric

Symmetric: A relation is symmetric if for all x,y ∈ X, (x,y) ∈ R ⇒ (y,x) ∈ R

Asymmetric: A relation is asymmetric if for all x,y ∈ X, (x,y) ∈ R ⇒ (y,x) ∉ R

Antisymmetric: A relation is antisymmetric if:

  • 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

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