Types of Relations

The Empty Relation between sets X and Y, or on E, is the empty set ∅

The Full Relation between sets X and Y is the set X × Y

The Identity Relation on set X is the set { (x, x) | x ∈ X }

The Inverse Relation R' of a relation R is defined as − R' = { (b, a) | (a, b) ∈ R }

Example − If R = { (1, 2), (2, 3) } then R' will be { (2, 1), (3, 2) }

A relation R on set A is called Reflexive if ∀ a ∈ A is related to a (aRa holds)

Example − The relation R = { (a, a), (b, b) } on set X = { a, b } is reflexive.

A relation R on a set A is called Irreflexive if no a ∈ A is related to an (aRa does not hold).

Example − The relation R = { (a, b), (b, a) } on set X = { a, b } is irreflexive.

A relation R on a set A is called Symmetric if xRy implies yRx, ∀ x ∈ A$ and ∀ y ∈ A.

Example − The relation R = { (1, 2), (2, 1), (3, 2), (2, 3) } on set A = { 1, 2, 3 } is symmetric.

A relation R on set A is called Anti-Symmetric if xRy and yRx implies x = y \: ∀ x ∈ A and ∀ y ∈ A.

Example − The relation R = { (x, y)→ N |x ≤ y } is anti-symmetric since x ≤ y and y ≤ x implies x = y.

A relation R on set A is called Transitive if xRy and yRz implies xRz, ∀ x,y,z ∈ A.

Example − The relation R = { (1, 2), (2, 3), (1, 3) } on set A = { 1, 2, 3 } is transitive.

A relation is an Equivalence Relation if it is reflexive, symmetric, and transitive.

Example − The relation 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 since it is reflexive, symmetric, and transitive.