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Mathematics LibreTexts

10.2: Properties of Functions

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Definition: Surjective Function

a function whose image is all of its codomain — that is, every element of the codomain is an output for the function;

Definition: Surjective Function

a surjective function

Definition: Onto

synonym for surjective

Definition: f:AB

function f is surjective

A function f:AB is surjective if f(A)=B. Since we have f(A)B by definition of image, to show that a function is surjective we only need to show f(A)B.

Test 10.2.1: Surjective function.

  • Function f:AB is surjective if Bf(A). That is, f is surjective if for every element bB, there exists at least one element aA such that f(a)=b.
  • Function f:AB is not surjective if there exists at least one element bB for which there is no element aA satisfying f(a)=b. (Equivalently, there exists bB for which every aA satisifes f(a)b.)

Definition: Injective Function

a function for which two different inputs never produce the same output

Definition: Injection

an injective function

Definition: Embedding

synonym for injection

Definition: One-to-one

synonym for injective

Definition: f:AB

function f is injective

Example 10.2.2: Demonstrating that a function is not injective.

The function f:RR, f(x)=x2, is not injective, since f has repeated outputs. For example, f(1)=f(1). And in fact, f(x)=f(x) for every xR.

Example 10.2.3: Demonstrating that a function is injective.

Verify that the function f:NN, f(n)=2n+1, is injective.

Solution

Using the contrapositive version of the Injective Function Test, suppose domain elements n1,n2N satisfy f(n1)=f(n2). Then using the formula defining the input-output rule for f, we have

2n1+1=2n2+1,
which reduces to n1=n2.

An injection f:AB gives us a way of thinking of A as a subset of B, by considering f(A)B.

Example 10.2.4: Turning letters into numbers.

Let Σ={a,b,,z}, and define φ:ΣN by the following table.

σ a b c z
φ(σ) 1 2 3 26

Then f embeds Σ into N in a familiar way, and lets us think of letters as numbers.

Definition: Bijective Function

a function that is both injective and surjective

Definition: Bijection

an bijective function

Definition: One-to-one Correspondence

synonym for bijection

Example 10.2.5

Function f:RR, f(x)=x3 is bijective.

A bijection f:AB allows us to think of A and B as essentially the same sets.

Example 10.2.6: Identifying letters with numbers.

Consider again f:ΣN from Example 10.2.4. If we write B=f(Σ)={1,2,3,,26}, then really we could think of the function as being defined f:ΣB. This version of f is bijective, and allows us to identify each letter with a corresponding number:

a1,b2,c3,,z26.
In this way, we can think of Σ and B as essentially the same set.

Example 10.2.7: Recognizing bijections.

Which of the following functions are bijections?

f:ZZ,g:ZN,h:ZZ,m2m,m|m|,mm.

Solution.

Is f bijective?.

No, f is not bijective because it is not surjective. For example, there is no mZ such that f(m)=1.

Is g bijective?.

No, g is not bijective because it is not injective. For example, g(1)=g(1).

Is h bijective?.

Yes, h is bijective. It is injective because if m1m2 then m1m2. And it is surjective because for nZ, we can realize n as an output n=h(m) by setting m=n.

Checkpoint 10.2.1: Bijections of counting sets.

For mN write

N<m={nNn<m={0,1,,m1}.
Prove that there exists a bijection N<lN<m if and only if =m.


This page titled 10.2: Properties of Functions is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.

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