Functions

Definition of a Function

Definition: A function is a rule that assigns to each element in a set D, exactly one element in a set E. Namely, set D refers to the domain of the function, while set E refers to the codomain of the function.

Injections (One-to-One Functions)

Definition: //A function, f, is one-to-one if for all x, y ∈ A, when x ≠ y, then f(x) ≠ f(y). Alternatively, when x = y, then f(x) = f(y), that is, for each x ∈ A, there is a unique value in that x is mapped to in the codomain. //

Surjections (Onto Functions)

Definition: A function, f, is onto if for all y ∈ B, then there exists an x ∈ A such that f(x) = y.

Bijections

Definition: A function, f, is a bijection if and only if it is both one-to-one and onto. Hence, a bijection is both an injection and a surjection.