Functions in Discrete Mathematics

These flashcards cover key vocabulary and concepts related to functions as outlined in the lecture notes.

Function

An assignment of exactly one element of a set to each element of another set.

Domain

The set of all inputs for a function.

Codomain

The set of all allowable outputs for a function.

Image

The output assigned to an input by a function.

Preimage

The input that corresponds to a given output in a function.

Many-One Function

A function where two or more elements of the domain have the same image in the codomain.

One-One Function (Injective Function)

A function where different elements of the domain are connected to different elements of the codomain.

Onto Function (Surjective Function)

A function where every element of the codomain is an image of some element of the domain.

Into Function

A function that is not onto, meaning at least one element in the codomain has no pre-image in the domain.

Inverse Function

A function that reverses the correspondence of the original function, denoted as f⁻¹.

Composition of Functions

The application of one function to the result of another function, denoted as (f ◦ g)(a) = f(g(a)).

Range

The set of all images of the elements of the domain.

Function Representation

Functions can be represented as f: A → B, where A is the domain and B is the codomain.

Example of Function

Assigning grades to students, where grades are determined based on student names.

Evaluation of Functions

Replacing the variable in a function with a value from the function's domain to compute the result.