TOPICS

Formulae

Relations and Functions

1. Reflexive, if (a, a) ∈ R, for every a ∈ A 2. Symmetric, if (a, b) ∈ R implies that (b, a) ∈ R, for every (a, b) ∈ A 3. Transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all (a, b, c) ∈ A. 4. Equivalence, if Reflexive, Symmetric, and Transitive Relation.

1. One-to-One or Injective, for every x₁, x₂ ∈ X, f(x₁) = f(x₂) implies x₁ = x₂. 2. On-to or Surjective, for every y ∈ Y, there exists an element x in X such that f(x) = y. 3. Bijetive, if Injective and Surjective function.

1. Composite, gof(x) = g(f(x)) 2. Composite, fog(x) = f(g(x)) 3. Composite, fof(x) = f(f(x)) 4. Invertible, f(x) = ax + b, then put f(x) = y, find value of x = ? i.e., f^{-1}(y) is a inverse value of x.

1. Commutative, if a + b = b + a and a x b = b x a for every a, b ∈ X. 2. Associative, if (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c) for every a, b, c ∈ X. 3. Identity, if e ∈ a then a ∗ e = a = e ∗ a, ∀ a ∈ A. 4. Invertible, if e ∈ a then a ∗ b = e = b ∗ a, ∀ a ∈ A i.e., a^{-1}

CLASSES