Types of Relations in Mathematics

Below, assume the relation R is on a set A (so both components come from the same set).

Reflexive Relation

Definition: R is reflexive if every element is related to itself: for all a ∈ A, (a, a) ∈ R.

Example: Equality “=” on any set is reflexive because a = a always.

Non-example: The “<” relation on numbers is not reflexive (no number is less than itself).

Irreflexive Relation

Definition: For all a ∈ A, (a, a) ∉ R. No element is related to itself.

Example: “<” on integers is irreflexive.

Symmetric Relation

Definition: R is symmetric if whenever (a, b) ∈ R, then (b, a) ∈ R.

Example: “is a sibling of” is symmetric; if A is sibling of B, then B is sibling of A.

Non-example: “≤” is not symmetric: a ≤ b does not imply b ≤ a unless a = b.

Antisymmetric Relation

Definition: R is antisymmetric if whenever (a, b) ∈ R and (b, a) ∈ R, then a = b.

Example: “≤” is antisymmetric: if a ≤ b and b ≤ a, then a = b.

Note: Antisymmetric ≠ “not symmetric.” A relation can be both symmetric and antisymmetric (e.g., equality).

Asymmetric Relation

Definition: R is asymmetric if whenever (a, b) ∈ R, then (b, a) ∉ R (and thus irreflexive).

Example: “<” is asymmetric: if a < b, then it is impossible that b < a.

Transitive Relation

Definition: R is transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.

Example: “≤” is transitive: from a ≤ b and b ≤ c, conclude a ≤ c.

Non-example: “is a friend of” is usually not transitive; A being friends with B and B with C doesn’t force A with C.

Equivalence Relation

Definition: A relation that is reflexive, symmetric, and transitive.

Behavior: Equivalence relations partition a set into equivalence classes of “mutually related” elements.

Example: “congruence modulo n” on integers: a ≡ b (mod n) iff n | (a − b).

Partial Order Relation

Definition: A relation that is reflexive, antisymmetric, and transitive. We call (A, ≤) a partially ordered set (poset).

Example: “⊆” (subset) on the power set of a set is a partial order.

Total Order: A partial order where every pair is comparable (for all a, b, either a ≤ b or b ≤ a).

Function (as a Special Relation)

Definition: A function is a relation where every element of the domain appears exactly once as a first component, mapping to a single second component.

Example: f = {(1, 4), (2, 5), (3, 6)} is a function; {(1, 4), (1, 5)} is not.

Inverse Relation

Definition: For a relation R, its inverse R⁻¹ is the set of reversed pairs: R<sup>−1</sup> = {(b, a) : (a, b) ∈ R}.

Properties: If R is symmetric, then R = R<sup>−1</sup>. If R is a function, R⁻¹ need not be a function.

Quick Tests and Intuition

• Reflexive? Check all self-pairs (a, a) are present.
• Irreflexive? Check no self-pair appears.
• Symmetric? Every time you see (a, b), confirm (b, a) is there.
• Antisymmetric? If you see both (a, b) and (b, a), ensure that forces a = b.
• Transitive? For each chain a R b and b R c, verify a R c.

Worked Mini-Example

Let A = {1, 2, 3} and define R = {(1,1), (1,2), (2,2), (2,3), (1,3), (3,3)}.

• Reflexive: Yes, (1,1), (2,2), (3,3) are present.
• Symmetric: No, (1,2) is in R but (2,1) is not.
• Antisymmetric: Yes, no distinct a≠b have both (a,b) and (b,a).
• Transitive: Yes, e.g., (1,2) and (2,3) imply (1,3), which is present.
• Conclusion: R is a partial order on A (reflexive, antisymmetric, transitive).