Engineering at Alberta Courses

Mathematical Preliminaries: Types of Maps

Learning Outcomes:

Differentiate between injective, surjective, and bijective maps

Types of Maps

Maps or functions between sets can be injective, surjective, bijective, or none of the above.

Injective Maps

An injective map also called “one-to-one” is a map that designates a unique image for every element. Let $T : X\rightarrow Y$ . $T$ is injective if $\forall x\in X : \exists! y\in Y$ such that $T(x) = y$ .

Surjective Maps

A surjective map also called “onto” is a map such that every element in the codomain has a pre-image. Let $T : X\rightarrow Y$ . $T$ is surjective if $\forall y\in Y: \exists x\in X$ such that $T(x) = y$ . In other words, there is always a pre-image for all the elements in $Y$ .

Bijective Maps

A bijective map also called “invertible” is a map such which is both injective and surjective. An bijective maps is a map that can be inverted, i.e., let $T : X\rightarrow Y$ . $T$ is bijective, then, there exists $G:Y\rightarrow X$ such that $G=T^{-1}$ . I.e., $G(T(x))=x$ .

Examples

For example, let $A=\mbox{set of students in CivE 398}$ and $B=\mbox{Set of IDs at the University of Alberta}$ . The map $f:A\rightarrow B$ assigns every student a unique ID in the set of IDs of students at the University of Alberta. This map is injective or one-to-one since each student has his/her own unique ID. However, this map is not surjective (onto) because there are IDs in $B$ that are not assigned to students in $A$ . Similarly, let $A=\mbox{set of students in CivE 398}$ and $B=\{male,female\}$ . There is a surjective map between the students in the class and the gender types as there are both male and female students taking CivE 398. Unless there is only one male student and one female student, this map is surjective but not injective. Figure 1 contains some illustrative examples of the above definitions.

Figure 1. Bijective, injective, and surjective maps

Video

Quiz 2 – Types of Maps

Solution Guide

Page Comments

Qisi says:

March 29, 2022 at 4:26 pm

Hi there, thank you very much for putting up this book. I’m a bit confused about the definition of injection in the text. For the case of f:R->R:f(x) = x^2, for any x in R, there is indeed a unique y s.t y=x^2. I’m having the feeling that the definition of injection should be something like:
any y in Y s.t. y=f(x), there exists a unique x in X that satisfies y=f(x),
or:
any x1,x2 in X, x1~=x2 => f(x1) ~= f(x2)

Reply
Samer Adeeb says:

March 29, 2022 at 7:26 pm

Hi Qisi,
The following two statements are equivalent:
– for all x1~=x2 => f(x1)~=f(x2)
– for all x => f(x) is unique.

Reply
audeser says:

June 6, 2023 at 5:11 am

Under “SURJECTIVE MAPS”, the “Let t : X\rightarrow Y” should be Let T : X\rightarrow Y”

Reply
1. Samer Adeeb says:
  
  June 6, 2023 at 9:08 am
  
  This has been correct. Thank you.
  
  Reply