Types of Relations and Functions PDF

Types of Relation Equivalence Relations Reflexive Relations (i) R is reflexive i.e. (a, a) ∈ R ∀ a ∈ A If (a, a) ∈ R, ∀ a ∈ R (ii) R is symmetric i.e. (a, b) ∈ R ⇒ (b, a) ∈ R (iii) R is transi...

Types of Relation Equivalence Relations Reflexive Relations (i) R is reflexive i.e. (a, a) ∈ R ∀ a ∈ A If (a, a) ∈ R, ∀ a ∈ R (ii) R is symmetric i.e. (a, b) ∈ R ⇒ (b, a) ∈ R (iii) R is transitive i.e. (a, b), (b, c) ∈ R ⇒ (a, c) ∈ Symmetric Relation R If(x, y) ∈ R ⇒ (y, x) ∈ R Types of Functions Identity functions : If f(x) = x, ∀ x ∈ A Anti Symmetric Relations If (a, b) ∈ R and (b, a) ∈ R ⇒ a = b One-One Functions (Injective) Transitive Relations f(a) = f(a’) ⇒ a = a’ If (a, b) ∈ R (b, a) ∈ R ⇒ (a, c) ∈ R Identity Relations If (a, b) ∈ R iff a = b Onto Functions (Surjective) Algebra of Functions Iff f(A) = B (f + g) (x) = f(x) + g(x) Range = Codomain (f - g) (x) = f(x) - g(x) (f - g) (x) = f(x). g(x) Many to one Function Into Function Where g(x) ≠ 0  Important Formulae If A and B are finite sets and O (A) = m, O (B) If O(A) = m, O(B) = n, then total number = n, m n.Then number of injection of mappings from A to B is nm (oneone) from A to B is If f : A 🠆 B is injective (one-one), then If f : A 🠆 B is surjective (onto), then O(A) ≤ O(B). O(A) ≥ O(B) If f : A 🠆 B is bijective (one-one onto), Let f : A 🠆 B and O(A) = )(B), then f is then O(A) = O(B). one-one ⇔ it is onto.