# What is Injective: Definition and 97 Discussions

In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.
A function f that is not injective is sometimes called many-to-one.

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1. ### ##G## is Injective ##\iff ## ## f ## is onto ## Y ## (Lambda Notation)

My attempt: ## ( \rightarrow ) ## Suppose G is injective. Let ## y \in Y ## be arbitrary, denote A = ## \{ y \} ## so that ## G(A) = G(\{ y \}) = f^{-1}[\{ y \}] = \{ x \in X | f(x) \in \{ y \} \} =\{ x \in X | f(x)= y \} ##. [ However, now I am stuck because I don't know if ## G(A)=...
2. ### I Proof involving functional graphs and the injective property

Definition: Let ##G## be a graph. ##G## is a functional graph if and only if ##(x_1,y_1) \in G## and ##(x_1,y_2) \in G## implies ##y_1=y_2##. Problem statement, as written: Let ##G## be a functional graph. Prove that ##G## is injective if and only if for arbitrary graphs ##J## and ##H##, ##G...
3. ### I Proof involving functional graphs and the injective property

My only qualm is that the statement “Let G be a functional graph” never came into play in my proof, although I believe it to be otherwise consistent. Can someone take a look and let me know if I missed something, please? Or is there another reason to include that piece of information?
4. ### I Is f(x) an Injective Function? Understanding Proof and Notation

I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...
5. ### Proving a function is injective

Hello, Let f: ]1, +inf[ → ]0, +inf[ be defined by f(x)=x^2 +2x +1. I am trying to prove f is injective. Let a,b be in ]1, +inf[ and suppose f(a) = f(b). Then, a^2 + 2a + 1 = b^2 + 2b + 1. How do I solve this equation such that I end up with a = b? Solution: (a + 1) ^2 = (b + 1)^2...
6. ### I Prove that if T is injective, T*T is invertible

I'm using the notation T* to indicate the adjoint of T. I got as far as to say that if T is injective, then T* is surjective. But I don't know how to show that T*T is invertible. Showing that T*T is surjective or injective would imply invertibility, but I'm not sure how to do that either. I...
7. ### Finding a useful denial of a injective function and a surjective function

Homework Statement Find the useful denial of a injective function and a surjective function. Homework EquationsThe Attempt at a Solution I know a one to one function is (∀x1,x2 ∈ X)(x1≠x2 ⇒ f(x1) ≠ f(x2)). So would the useful denial be (∃x1,x2 ∈ X)(x1 ≠ x2 ∧ f(x1) = f(x2))? I know a onto...
8. ### MHB Are Injective R-Linear Mappings in C Necessarily Surjective?

I am reading Reinhold Remmert's book "Theory of Complex Functions" ... I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.4: Angle-Preserving Mappings ... ... I need help in order to fully understand a remark of Remmert's regarding...
9. ### I Injective immersion that is not a smooth embedding

Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8 ##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)## As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...
10. ### MHB Monoids .... Monos and Injective Maps .... Awodey, Example 2.3, Page 30 ....

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Chapter 2: Abstract Structures ... ... I need some help in order to fully understand Awodey Example 2.3, Chapter 2 ... ... Awodey Example 2.3, Chapter 2 reads as follows: In the Example above Awodey writes the...
11. ### Find injective homomorphism from D_2n to S_n

Homework Statement Find, with justification, an injective group homomorphism from ##D_{2n}## into ##S_n##. Homework EquationsThe Attempt at a Solution So I'm thinking that the idea is to map ##r## and ##s## to elements in ##S_n## that obey the same relations that r and s satisfy. I can see how...
12. ### A Is the Injective Hull of an Irreducible Module in Group K?

Let's suppose that I have an element ##e## of order ##p## in the group of complex numbers whose elements all have order ##p^n## for some ##n\in\mathbb{N}## (henceforth called ##K##), and the module generated by ##(e)## is irreducible. How do I show that the injective hull of the module...
13. ### I Derivative of this function is injective everywhere

I'm reading a pdf where it's said that the function ##f: \mathbb R \longrightarrow \mathbb{R}^2## given by ##f(x) = \langle \sin (2 \pi x), \cos ( 2 \pi x) \rangle## is not one-to-one, because ##f(x+1) = f(x)##. This is pretty obvious to me. What I don't understand is that next they say that the...
14. ### MHB Proving Injectivity of a Vector-Valued Function Using the Mean Value Theorem

Hey! :o I want to prove the following criteroin using the mean value theorem for differential calculus in $\mathbb{R}^n$: Let $G\subset \mathbb{R}^n$ a convex region, $f:G\rightarrow \mathbb{R}^n$ continuously differentiable and it holds that \begin{equation*}\det...
15. ### MHB Prove that if g(f(x)) is injective then f is injective

Dear Everybody, Question: "Prove that if g(f(x)) is injective then f is injective" Work: Proof: Suppose g(f(x)) is injective. Then g(f(x1))=g(f(x2)) for some x1,x2 belongs to C implies that x1=x2. Let y1 and y2 belongs to C. Since g is a function, then y1=y2 implies that g(y1)=g(y2). Suppose...
16. S

### B How Can You Determine if an Operator is Surjective, Injective, or Bijective?

Hi, I found in Kreyszig that if for any ##x_1\ and\ x_2\ \in \mathscr{D}(T)## then an injective operator gives: ##x_1 \ne x_2 \rightarrow Tx_1 \ne Tx_2 ## and ##x_1 = x_2 \rightarrow Tx_1 = Tx_2 ##If one has an operator T, is there an inequality or equality one can deduce from this, in...
17. ### How to check if a transformation is surjective and injective

Homework Statement I have attached the question. Translated: Suppose T: R^4 -> R^4 is the image so that: ... Homework Equations So I did this question and my final answers were correct: 1. not surjective 2. not injective. My method of solving this question is completely different than the...
18. ### I Does this theorem need that Ker{F}=0?

I have encountered this theorem in Serge Lang's linear algebra: Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W. In the proof he starts with C1F(v1) +...
19. ### Is there an injective ring homomorphism?

Homework Statement From ##\mathbb{Z}_3## to ##\mathbb{Z}_{15}## Homework EquationsThe Attempt at a Solution I know how to do this if we assumed that the rings had to be unital. In that case, there can be no non-trivial homomorphism. However, in my book rings don't need unity, and so a...
20. ### Injective & Surjective Functions

Just wondering if anyone could help me get in the right direction with these questions and/or point me to some material that will help me better understand how to approach these questions In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R. (a) Show that a function...
21. ### I Proving Injectivity and Surjectivity: A Fundamental Concept in Function Theory

Stumped on a couple of questions, if anyone could help! In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R. (a) Show that a function f is surjective if and only if there exists a function g such that f ◦ g = I. (b) Show that a function f is injective if and only if...
22. ### I DimKer(T) = 0 <--> T is injective

Let T be linear transformation from V to W. I know how to prove the result that nullity(T) = 0 if and only if T is an injective linear transformation. Sketch of proof: If nullity(T) = 0, then ker(T) = {0}. So T(x) = T(y) --> T(x) - T(y) = 0 --> T(x-y) = 0 --> x-y = 0 --> x = y, which shows that...
23. ### Proving local injectivity of curve

Homework Statement Let γ : I → Rn be a regular smooth curve. Show that the map γ is locally injective, that is for all t0 ∈ I there is some ε > 0 so that γ is injective when restricted to (t0 − ε , t0 + ε ) ∩ I. Homework Equations The Attempt at a Solution [/B] So I know a function (or a...
24. ### Prove that an endomorphism is injective iff it is surjective

Homework Statement Prove that an endomorphism between two finite sets is injective iff it is surjective Homework EquationsThe Attempt at a Solution I can explain this in words. First assume that it is injective. This means that every element in the domain is mapped to a single, unique element...
25. ### MHB Injective linear mapping - image

Hey! :o Let $F$ be a field and $V,W$ finite-dimensional vector spaces over $F$. Let $f:V\rightarrow W$ a $F$-linear mapping. We have to show that $f$ is injective if and only if for each linearly independent subset $S$ of $V$ the Image $f(S)$ is linearly independent in $W$. I have done the...
26. ### MHB Determining if a function is injective

is the function x³-5x²+3x+5 injective. how can you tell
27. ### I Multivariable function that is injective?

Hey all, is it possible to find a function that for $$a,b,c.. \in \mathbb{R}$$ $$y= f(a,b,c,..) , \hspace{5mm} y= \rho , \rho \in \mathbb{R} \hspace{2mm} for \hspace{2mm} only \hspace{2mm} 1 \hspace{2mm} set \hspace{2mm} of \hspace{2mm} a,b,c..$$ Any help appreciated
28. ### I Symmetric injective mapping from N² to N

Hi, I've been trying to find one symmetric "injective" N²->N function, but could not find any. The quotes are there because the function I'm trying to find is not really injective, as I need that the two arguments be interchangeable and the value remains the same. In other words, the tuple (a...
29. ### Show a functions inverse is injective iff f is surjective

Hello all, Can anyone give me a pointer on how to start this proof?: f:E\rightarrow F we consider f^{-1} as a function from P(F) to P(E). Show f^(-1) is injective iff f is surjective.
30. ### Set Theory: Prove a function is injective

Homework Statement Hello, I need some help on the following. I am BRAND new to set theory and this was in my first HW assignment and I am not sure where to start. The question is as follows: Let A and B be parts of a set E Let P(E)\rightarrow P(A) X P(B) be defined by f(X)=(A\cap X,B\cap X)...
31. ### Proving Left Inverse of Injective Function

Homework Statement ##f : A \rightarrow B## if and only if ##\exists g : B \rightarrow A## with the property ##(g \circ f)(a) = a##, for all ##a \in A## (In other words, ##g## is the left inverse of ##f##) Homework EquationsThe Attempt at a Solution I have already prove the one direction. Now I...
32. ### Proving sinx+cosx is not one-one in [0,π/2]

Homework Statement Prove that sinx+cosx is not one-one in [0,π/2] Homework Equations None The Attempt at a Solution Let f(α)=f(β) Then sinα+cosα=sinβ+cosβ => √2sin(α+π/4)=√2sin(β+π/4) => α=β so it has to be one-one [/B]
33. ### MHB Injective and surjective functions

Hello, I've been reading about injectivity from Z to N and surjectivity from N to Z and was wondering whether there was some kind of algorithm that could generate these specific types of functions?
34. ### MHB Prove by induction that the function is injective

Hi! (Wave) The set $\mathbb{R}$ of real numbers is not countable. Proof: We define the function $F: \{0,1\}^{\omega} \to \mathbb{R}$ with the formula: $$(a_n)_{n \in \omega} \in \{0,1\}^{\omega} \mapsto F((a_n)_{n \in \omega})=\sum_{n=0}^{\infty} \frac{2a_n}{3^{n+1}}$$ Show that $F$ is 1-1...
35. ### Verify that the function is a injective

Homework Statement The function is ##\phi " \mathbb{Z}_{12} \rightarrow \mathbb{Z}_{24}##, where the rule is ##\phi ([a]_{12}) = [2a]_{24}##. Verify this is a injection Homework EquationsThe Attempt at a Solution Let ##[x]_{12} ,[y]_{12} \in \mathbb{Z}_{12}## be arbitrary. Suppose that...
36. ### What is is neither injective, surjective, and bijective?

As the title says.
37. ### MHB Functions that are injective but not surjective

I am reading Paolo Aluffi's book Algebra: CHapter 0. In Chapter 1, Section 2: Fumctions between sets we find the following: (see page 13) "if a function is injective but not surjective, then it will necessarily have more than one left-inverse ... " Can anyone demonstrate why this is true...
38. ### Show F is Injective & Cardinality of Domain

Homework Statement Let ## S = \{ (m,n) : m,n \in \mathbb{N} \} \\ ## a.) Show function ## f: S -> \mathbb{N} ## defined by ## f(m,n) = 2^m 3^n ## is injective b.) Use part a.) to show cardinality of S. The Attempt at a Solution a.) ## f(a,b) = f(c, d ) ; a,b,c,d \in \mathbb{N} \\\\ 2^a...
39. ### Demonstrating that a mapping is injective

I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules. I am studying Proposition 28 (D&F pages 387 - 388) In the latter stages of the proof of Proposition 28 we find the following statement (top of page 388): "In general, Hom_R (R, X) \cong X...
40. ### MHB Demonstrating that a mapping is injective

I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules. I am studying Proposition 28 (D&F pages 387 - 388) In the latter stages of the proof of Proposition 28 we find the following statement (top of page 388): "In general, Hom_R (R, X) \cong X...
41. ### Injective endomorphism = Surjective endomorphism

Is an injective endomorphism necessarily surjective? And it is also true the opposite?
42. ### MHB If Derivative is Not Zero Anywhere Then Function is Injective.

Hello MHB. I am sorry that I haven't been able to take part in discussions lately because I have been really busy. I am having trouble with a question. In a past year paper of an exam I am preparing for it read: Let $f: (a,b)\to \mathbb R$ be a differentiable function with $f'(x)\neq 0$ for...
43. ### Show that if f: A → B is injective and E is a subset of A, then f −1(f

Homework Statement Show that if f: A → B is injective and E is a subset of A, then f −1(f(E) = E Homework Equations The Attempt at a Solution Let x be in E. This implies that f(x) is in f(E). Since f is injective, it has an inverse. Applying the inverse function we see that...
44. ### Conditions for Surjective and Injective linear maps

Hello, I'm not sure if this should go under the HW/CW section, since it's not really a homework question, just a curiosity about certain kinds of functions. My specific question is this: If M: U→V is injective and dim(U)=dim(V), does that imply that M is surjective (and therefore...
45. ### Why is this function injective?

Homework Statement The function from R to R satisfies x + f(x) = f(f(x)) Find all Solutions of the equation f(f(x)) = 0. Part of the problem solution says that if f(x) = f(y), then "obviously" x = y. I understand the rest of the solution, but why does f(x) = f(y) imply that x = y?
46. ### Showing a binary operation is injective

Homework Statement The objective was to think of a binary operation ##*:\mathbb{N}\times\mathbb{N}\to\mathbb{N}## that is injective. A classmate came up with the following operation, but had trouble showing it was injective: ##a*b=a^3+b^4##. Homework Equations The Attempt at a...
47. ### MHB Surjective and injective linear map

I quote an unsolved question from MHF posted by user jackGee on February 3rd, 2013. P.S. Of course, I meant in the title and instead of an.
48. ### Show Injectivity of Mapping y=Ax for Finite Alphabet w/o Search Algorithm

Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. How can i show if the mapping from x to y is injective for given A and alphabet (beside a search method)? For example, let A and the alphabet be [1 0 1/sqrt2 1/sqrt2] [0 1 1/sqrt2 -1/sqrt2] and...
49. ### Proving Existence of Injective Binary Operation on $\mathbb{N}\times\mathbb{N}$

Homework Statement Prove or disprove: \exists a binary operation *:\mathbb{N}\times\mathbb{N}\to\mathbb{N} that is injective. Homework EquationsThe Attempt at a Solution At first, I was under the impression that I could prove this using the following operation. I define * to be...
50. ### Condition for a function to be injective

Homework Statement Do all the preimages on X need to have a (and of course I know only one but) image in Y for the f:x->y to be injective? IS THE FOLLOWING FUNCTION INJECTIVE SINCE ONE ELEMENT OF FIRST DOES NOT HAVE ANY IMAGE Homework Equations The Attempt at a Solution Thank You.