Relations and Functions

A × B = {(a, b) : a ∈ A, b ∈ B}

n(A × B) = n(A) × n(B) = p × q

A × A × A = {(a, b, c) : a, b, c ∈ A}

R ⊆ A × B

Total number of relations from A to B = 2^(pq)

f: A → B such that ∀ a ∈ A, ∃ unique b ∈ B with f(a) = b

Range of f = {f(x) : x ∈ A} ⊆ B

(f + g)(x) = f(x) + g(x)

(f − g)(x) = f(x) − g(x)

(αf)(x) = α · f(x)

(fg)(x) = f(x) · g(x)

(f/g)(x) = f(x)/g(x), provided g(x) ≠ 0