MATH 419 - Big Idea 1: Bijections

Flashcards covering definitions and theorems related to functions and bijections.

Function from A to B

A rule that assigns to each element of A a single element of B.

Domain of f

The set A in the function f: A → B.

Codomain of f

The set B in the function f: A → B.

Image of C

The set of all elements in B that are paired with an element of C, where C ⊂ A.

Image of f

Im(A).

Identity function

The function iA: A → A given by i(a) = a for all a ∈ A.

One-to-one / Injective function

No two elements of the domain have the same image; i.e., f(a1) = f(a2) implies a1 = a2.

Onto / Surjective function

Every element of B is the image of an element in A; i.e., for all b ∈ B, there exists a ∈ A such that f(a) = b.

Bijection

A function that is both one-to-one and onto.

Composition of g with f (g ◦ f)

Given by (g ◦ f)(a) = g(f(a)).

Invertible function f: A → B

There exists a function g: B → A such that g ◦ f = iA and f ◦ g = iB. Here, g is called the inverse of f.

Theorem: Invertible Function and Bijection

A function f : A → B is invertible if and only if f is a bijection.