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Types of Relations
A relation on a set can have various properties that classify it into different types. Understanding these types is essential for studying equivalence classes, partial orders, and other structures in discrete mathematics.
Empty Relation
The empty relation between sets X and Y, or on a set E, is the empty set ∅. No element is related to any other element.
Full Relation
The full relation (or universal relation) between sets X and Y is the entire Cartesian product X × Y. Every element in X is related to every element in Y.
Identity Relation
The identity relation on set X is the set { (x, x) | x ∈ X }. Every element is related only to itself.
Inverse Relation
The inverse relation R' of a relation R is defined as R' = { (b, a) | (a, b) ∈ R }. It reverses the direction of every ordered pair.
Example − If R = { (1, 2), (2, 3) } then R' = { (2, 1), (3, 2) }.
Reflexive Relation
A relation R on set A is called reflexive if every element is related to itself − that is, (a, a) ∈ R for all a ∈ A.
Example − R = { (a, a), (b, b) } on set X = { a, b } is reflexive.
Irreflexive Relation
A relation R on set A is called irreflexive if no element is related to itself − that is, (a, a) ∉ R for all a ∈ A.
Example − R = { (a, b), (b, a) } on set X = { a, b } is irreflexive.
Symmetric Relation
A relation R on set A is called symmetric if xRy implies yRx, for all x, y ∈ A.
Example − R = { (1, 2), (2, 1), (3, 2), (2, 3) } on set A = { 1, 2, 3 } is symmetric.
Anti-Symmetric Relation
A relation R on set A is called anti-symmetric if xRy and yRx together imply x = y, for all x, y ∈ A. Two-way edges are only allowed as self-loops.
Example − The relation "≤" on natural numbers is anti-symmetric, since x ≤ y and y ≤ x implies x = y.
Transitive Relation
A relation R on set A is called transitive if xRy and yRz implies xRz, for all x, y, z ∈ A.
Example − R = { (1, 2), (2, 3), (1, 3) } on set A = { 1, 2, 3 } is transitive.
Equivalence Relation
A relation is an equivalence relation if it is reflexive, symmetric, and transitive simultaneously.
Example − R = { (1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2), (1,3), (3,1) } on set A = { 1, 2, 3 } is an equivalence relation.
Summary Table
| Type | Condition | Graph Clue |
|---|---|---|
| Reflexive | (a, a) ∈ R for all a | Self-loop on every vertex |
| Irreflexive | (a, a) ∉ R for all a | No self-loops |
| Symmetric | (a, b) ∈ R ⇒ (b, a) ∈ R | Every edge has a reverse |
| Anti-symmetric | (a, b) ∈ R and (b, a) ∈ R ⇒ a = b | No two-way edges (except loops) |
| Transitive | (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R | Shortcut edge always exists |
| Equivalence | Reflexive + Symmetric + Transitive | All three properties visible |
Conclusion
Relations are classified by properties like reflexivity, symmetry, and transitivity. An equivalence relation combines all three properties and partitions a set into equivalence classes where all elements within a class are related to each other.
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