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24. Functions

The chapter focuses on the concept of functions in mathematics, encompassing various types of functions including injective, surjective, and bijective functions. It explains the fundamental characteristics of functions, such as their domain and co-domain, as well as the concepts of function composition and inverse functions. A detailed exploration of these topics aids in understanding their applications in discrete mathematics.

Sections

Functions

This section introduces the concept of functions, including definitions, types such as injective, surjective, and bijective functions, and the operation of composing functions.

24.1 Section Overview

Start current section content and materials

24.1.1 Definition of Function

A function establishes a unique relationship between elements of two sets, where each element of the first set maps to exactly one element in the second set.

24.1.2 Injective Functions

This section provides an overview of injective functions, their definitions, and properties, including discussions on domains and co-domains.

24.1.3 Surjective Functions

This section introduces surjective functions, discussing their characteristics and distinguishing them from other types of functions.

24.1.4 Bijective Functions

This section introduces bijective functions, which are both injective and surjective, ensuring a one-to-one correspondence between two sets.

24.1.5 Inverse of a Function

This section explores the concept of inverse functions, explaining how a function needs to be a bijection to have an inverse.

24.1.6 Composition of Functions

This section explains the concept and significance of function composition in mathematics, detailing its prerequisites and implications.

Learning Objectives

  • A function is a specific type of relation wherein each element from the domain is associated with exactly one element from the co-domain.

  • Functions can be classified into various categories based on their properties: injective (one-to-one), surjective (onto), and bijective (both one-to-one and onto).

  • The composition of functions is defined only when the range of the first function is a subset of the domain of the second function, and it is not necessarily commutative.

Key Concepts

Function

A function is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.

Injective Function

A function is injective if distinct elements in the domain map to distinct elements in the co-domain.

Surjective Function

A function is surjective if every element of the co-domain is mapped to by at least one element from the domain.

Bijective Function

A function is bijective if it is both injective and surjective, meaning there is a one-to-one correspondence between elements of the domain and co-domain.

Composition of Functions

The composition of two functions is a function that applies one function to the result of another function.

Inverse Function

An inverse function reverses the mapping of the original function, and it exists only if the function is a bijection.

Practice Exercises

Total Questions

2

Estimated Time

4 min

Passing Score

70%

Instructions

  • Read each question carefully
  • You can use hints if you need help
  • Complete all questions before submitting
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