What is Surjective: Definition and 86 Discussions

In mathematics, a function f from a set X to a set Y is surjective (also known as onto, or a surjection), if for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that f(x) = y. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word sur means over or above, and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.
Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection.

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  1. L

    I Create a surjective function from [0,1]^n→S^n

    the first method is this : I think I can create a surjective function f:[0,1]^n→S^n in this way : [0,1]^n is omeomorphic to D^n and D^n/S^1 is omeomorphic to S^n so finding a surjective map f is equal to finding a surjective map f':D^n →D^n/S^n and that is quotient map. Now if I take now a...
  2. M

    Showing that a function is surjective onto a set

    I have to show that $\forall z\in B(0,0.4)$, there exists an $x\in B(0,1)$ such that $f(x)=z$ but I am not sure how to show this. From the reverse triangle inequality $$-|f(x)-f(y)|+|x-y|\leq 0.1|x-y|\implies |f(x)-f(y)|\geq 0.9|x-y|$$ im not sure if this helps.
  3. JD_PM

    I Find the parameter values that make ##L## surjective

    I am given the following matrix representation of a linear mapping $$L_{\alpha}^{\beta}= \begin{pmatrix} 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & a & 0 & b & 0 & c \\ a & 0 & b & 0 & c & 0 \\ 0 & a^3 & 0 & b^3 & 0 & c^3 \\ a^3 & 0 & b^3 & 0 & c^3 & 0 \\...
  4. V

    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...
  5. S

    B Can a function become surjective by restricting its codomain?

    wikipedia says: "The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers R+, then g becomes bijective; its inverse is the...
  6. S

    MHB A question about surjective module-homomorphisms

    I have the following question about surjective module-homomorphisms. Let $f:A \longrightarrow B$ be a surjective $R$-homomorphism between $R$-modules $A$ and $B$. Let $S, T$ be submodules of $A$ and let $X, Y$ be submodules of $B$. I can prove that in general $$f(S+T)=f(S)+f(T)$$ and in...
  7. Math Amateur

    MHB Injective R-Linear Mappings in C are Surjective .... Remmert Ch. 0, Section 1.4 ....

    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...
  8. S

    B Surjective/injective operators

    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...
  9. C

    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...
  10. T

    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...
  11. T

    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...
  12. Mr Davis 97

    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...
  13. E

    MHB A Surjective function from [0,1]\{1/2} to [0,1]

    Hello guys! I'm taking Discrete Mathematics this semester and I got this question in one of my homework tasks. I've tried thinking about the solution over and over but can't seem to come up with anything.. The question goes like this: Is there a Surjective function from [0,1]\{1/2} to [0,1] such...
  14. M

    MHB Show that the mapping is surjective

    Hey! :o I am looking at the following exercise: Let $C$ be an algebraic closure of $F$, let $f\in F[x]$ be irreducible and let $a,b\in C$ be roots of $f$. Applying the theorem: "If $E$ is an algebraic extension of $F$, $C$ is an algebraic closure of $F$, and $i$ is an embedding (that is, a...
  15. R

    MHB Proving A Mapping is Surjective

    I have a mapping $L: \mathbb R^3 \rightarrow \mathbb R^3$ as defined by $L(x, y, z) = (x+z, y+z, x+y).$ How do you prove that the $L$ is an onto mapping? I know for sure that $\forall x, y, z \in \mathbb R$, then $x+z, y+z, x+y \in \mathbb R$ too. Then I need to prove that $Im (L) = \mathbb...
  16. squelch

    How many surjective functions are there from {1,2,...,n} to {a,b,c,d}?

    Homework Statement Count the number of surjective functions from {1,2,...,n} to {a,b,c,d}. Use a formula derived from the following four-set venn diagram: Homework Equations None provided. The Attempt at a Solution First, I divided the Venn diagram into sets A,B,C,D and tried to express...
  17. P

    Show that if ##f:A\rightarrow B## is surjective then....

    Homework Statement Show that if ##f:A\rightarrow B## is subjective and ##H\subseteq B## then ##f(f^{-1}(H))=H##, give an example to show the equality need to hold if ##f## if not surjective. Homework Equations 3. The Attempt at a Solution [/B] I know that I want to show that an element...
  18. DeldotB

    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.
  19. K

    Why does an inverse exist only for surjective functions?

    In other words, why does a function have to be onto (or surjective, i.e. Range=Codomain) for its inverse to exist?
  20. C

    MHB Surjective functions from a set of size n+3 to a size of n

    Hello, I wonder if anyone could settle a disagreement I'm having with one of my peers. The question is 'How many surjective functions are there from a set of size n+3 to a set of size n?'. Now, I've already proven that there are (n+1 choose 2)n! surjective functions from a set of size n+1 to a...
  21. C

    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?
  22. P

    Proving a cubic is surjective.

    Homework Statement $$f:\mathbb{R}\rightarrow\mathbb{R}~~\text{where}~~f(x)=x^3+2x^2-x+1$$ Show if f is injective, surjective or bijective. Homework EquationsThe Attempt at a Solution f is obviously not injective (and thus not bijective), one counter example is x=-1 and x=1. I can see from the...
  23. evinda

    MHB How can we show that h is surjective?

    Hello! (Wave) I am looking at the proof of the following proposition: The union of two finite sets is a finite set.Proof: Let $X,Y$ finite sets. Then there are $n,m \in \omega$ such that $X \sim n$ and $Y \sim m$, i.e. there are $f: X \overset{\text{1-1 & surjective}}{\longrightarrow}n, g: Y...
  24. W

    What is is neither injective, surjective, and bijective?

    As the title says.
  25. DavideGenoa

    Existence of surjective linear operator

    Dear friends, I read that, if ##A## is a bounded linear operator transforming -I think that such a terminology implies that ##A## is surjective because if ##B=A## and ##A## weren't surjective, that would be a counterexample to the theorem; please correct me if I'm wrong- a Banach space ##E##...
  26. K

    Composition of surjective functions

    I'm learning maths myself, but I'm going to university in 2 months. This is my first try at proving anything. Homework Statement Prove that the composition of surjective functions is also a surjection. Homework Equations A definition of surjective function: If f:S_1\rightarrow S_2...
  27. Math Amateur

    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...
  28. A

    Surjective Proof Homework: Show f is Surjective on (c,d)

    Homework Statement Suppose f: (a,b)→R where (a,b)\subsetR is an open interval and f is a differentiable function. Assume that f'(x)≠0 for all x\in(a,b). Show that there is an open interval (c,d)\subsetR such that f[(a,b)]=(c,d), i.e. f is surjective on (c,d). Homework Equations f is...
  29. B

    Prove that a function is surjective

    Homework Statement Let PosZ = {z ∈ Z : z > 0}. Consider the function f : PosZ → PosZ dened as follows: • f(1) = 1 • If z ∈ PosZ and z > 1 then f(z) is the largest integer that divides z but is distinct from z. (For example f(41) = 1 and f(36) = 18.) Prove that f is surjective...
  30. Daaavde

    Injective endomorphism = Surjective endomorphism

    Is an injective endomorphism necessarily surjective? And it is also true the opposite?
  31. M

    Surjective function g and the floor function

    Homework Statement . Let ##A## be the set of sequences ##\{a_n\}_{n \in \mathbb N}##: 1) ##a_n \in \mathbb N## 2) ##a_n<a_{n+1}## 3) ##\lim_{n \to \infty} \frac {\sharp\{j: a_j \leq n\}} {n}## exists.Call that limit ##\delta (a_n)## and define the distance (I've already proved this is a...
  32. S

    Surjective proof & finding inverse

    prove the function ## g: \mathbb{N} \rightarrow \mathbb{N} ## ## g(x) = \left[\dfrac{3x+1}{3} \right] ## where ## [y] ## is the maximum integer part of r belonging to integers s.t. r less than or equal to y is surjective and find it's inverse I know this function is bijective, but how do I...
  33. S

    Proving functions are surjective

    prove whether or not the following functions are surjective or injective: 1) g: \mathbb{R} \rightarrow \mathbb{R} g(x) = 3x^3 - 2x 2) g: \mathbb{Z} \rightarrow \mathbb{Z} g(x) = 3x^3 - 2x my working for 1): injective: suppose g(x') = g(x) : 3x'^3 - 2x' = 3x^3 - 2x this does not imply...
  34. M

    A separable metric space and surjective, continuous function

    Homework Statement . Let X, Y be metric spaces and ##f:X→Y## a continuous and surjective function. Prove that if X is separable then Y is separable. The attempt at a solution. I've tried to show separabilty of Y by exhibiting explicitly a dense enumerable subset of Y: X is separable...
  35. S

    Show that if f: A→B is surjective and and H is a subset of B, then f([

    [f]^{}[/2]Homework Statement Show that if f: A→B is surjective and and H is a subset of B, then f(f^(-1)(H)) = H. Homework Equations The Attempt at a Solution Let y be an element of f(f^(-1)(H)). Since f is surjective, there exists an element x in f^(-1)(H) such that f(x) =...
  36. M

    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...
  37. E

    Let f:G -> H be a surjective homomorphism. |C_G(g)| >= |C_H(f(g)|

    Let f:G --> H be a surjective homomorphism. |C_G(g)| >= |C_H(f(g)| Homework Statement Suppose G is a finite group and H is a group, where θ:G→H is a surjective homomorphism. Let g be in G. Show that |CG(g)| ≥ |CH(θ(g))|. Homework Equations This problem has been bugging me for a day now. I'm...
  38. Fernando Revilla

    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.
  39. S

    Determining if a function is surjective

    I understand the concept of a surjective or onto function (to a degree). I understand that if the range and domain of the function are the same then the function is onto. My professor gave an additional definition which I did not understand. Here it goes: \forally\inB \existsx\inA...
  40. L

    Proving a surjective map iff the map of the inverse image is itself

    In the recommended format :) Homework Statement First we say that f:S→T is a map. If Y ⊆ T and we define f-1(Y) to be the largest subset of S which f maps to Y: f-1(Y) = {x:x ∈ S and f(x) ∈ Y} I must prove that f[f-1(Y)] = Y for every subset Y of T if, and only if, T = f(S). Homework...
  41. A

    Prove that f is surjective iff f has a right inverse. (Axiom of choice)

    Homework Statement Suppose f: A → B is a function. Show that f is surjective if and only if there exists g: B→A such that fog=iB, where i is the identity function.The Attempt at a Solution Well, I believe for a rigorous proof we need to use the axiom of choice, but because I have never worked...
  42. S

    Proof: Permutations and Surjective Functions

    Homework Statement Let X and Y be finite nonempty sets, |X|=m, |Y|=n≤m. Let f(n, m) denote the number of partitions of X into n subsets. Prove that the number of surjective functions X→Y is n!*f(n,m). Homework Equations I know a function is onto if and only if every element of Y is mapped...
  43. P

    Show that linear transformation is surjective but not injective

    Hi, My question is to show that the linear transformation T: M2x2(F) -> P2(F) defined by T (a b c d) = (a-d) | (b-d)x | (c-d)x2 is surjective but not injective. thanks in advance.
  44. A

    Injective and Surjective linear transformations

    I was struck with the following question: Is there a linear map that's injective, but not surjective? I know full well the difference between the concepts, but I'll explain why I have this question. Given two finite spaces V and W and a transformation T: V→W represented by a matrix \textbf{A}...
  45. R

    Proving Surjectivity of Mapping A_g:G ---> G for Automorphism

    I have a mapping A_g:G ---> G defined by A_g(x) = g^-1(x)g (for all x in G) and as part of showing it is an automorphism i need show it is surjective. I'm not entirely sure how to do this but have made an attempt and would appreciate and feedback or hints to what I actually need to...
  46. A

    Denumerable set and surjective function

    This is the problem: "Prove that if A is denumerable and there exists a g: A -> B that is surjective, then there exists an h: B -> A so that h is injective." So I've started it as: Suppose a set A is denumerable and a function f: A-> B is surjective. Since there exists a surjective...
  47. S

    Surjective functions and partitions.

    Let A be the set of all functions f:{1,2,3,4,5}->{1,2,3} and for i=1,2,3 let Ai denote a subset of the functions f:{1,2,3,4,5}->{1,2,3}\i. i)What is the size of : 1). A, 2).the sizes of its subsets Ai,and 3).Ai\capAj (i<j) also 4).A1\capA2\capA3. ii)Find with justification the...
  48. B

    Counting Surjective Maps from a Finite Set to Itself

    Homework Statement Let S = {1,2,3,...,n} How many surjective maps are there from S to S? Homework Equations n/a The Attempt at a Solution The book's answer is n! However, I thought that total number of surjective maps = n^n because 1-1 isn't required. Where am I wrong?
  49. S

    Understanding the Surjectivity of the Norm Function in Finite Fields

    If we have N:F_q^n ...> F_q , be the norm function . can anyone explian how the map N is surjective .
  50. G

    Composition of Mappings, Surjective and Injective

    Homework Statement a) Let g: A => B, and f: B => C. Prove that f is one-to-one if f o g is one-to-one. b) Let g: A => B, and f: B => C. Prove that f is onto if f o g is onto. Homework Equations a) Since f o g is onto, then (f o g)(a) = (f o g)(b) => a = b. b) Since f o g is onto, every element...
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