Relation

For any non-empty sets A and B, a subset of A × B is called a relation from A to B.

If S is a relation in A i.e. S ⊂ A × A and (x, y) ∈ S, we say x is related to y by S or xSy.

Types of Relation

Void or Empty Relation

A relation in the set A with no elements is called an empty relation. Φ ⊂ A × A, Φ is a relation called empty relation.

Universal Relation

A relation in the set A which is A × A itself is called a universal relation.

Reflexive Relation

If S is a relation in the set A and aSa, ∀ a ∈ A i.e. (a, a) ∈ S, a ∈ A, we say S is a reflexive relation.

Symmetric Relation

If S is a relation in a set A and if aSb ⇒ bSa i.e. (a, b) ∈ S ⇒ (b, a) ∈ S ∀ a, b ∈ A. We say S is a symmetric relation in A.

Transitive Relation

If S is a relation in the set A and if aSb and bSc ⇒ aSc ∀ a, b, c ∈ A i.e. (a, b) ∈ S and (b, c) ∈ S ⇒ (a, c) ∈ S ∀ a, b, c ∈ A, thus we say that S is a transitive relation in A.

Equivalence Relation

If a relation S in a set A is reflexive, symmetric and transitive is called an equivalence relation in A.

If S is equivalence relation and (x, y) ∈ S then x ~ y.

Anti symmetric Relation

If S is a relation in A and if (a, b) ∈ S and (b, a) ∈ S ⇒ a = b ∀ a, b, ∈ A, then S is said be an anti-symmetric relation.

Equivalence Classes

Let A be an equivalence relation in A. let a ∈ A, then the subset {x ∈ A, xSa} is said to be equivalence class corresponding to a.

Points To Remember

  • If A has m and B has n elements, then A × B has mn odered pairs
  • A relation R is a set is said to be identify relation if R = {(a, a); a ∈ A}
  • Identify relation on a non-empty set is an equivalence relation.
  • Universal relation on a non-empty set is an universal relation.
  • Identify relation on a non-empty set is anti-symmetric.