# 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. ### 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. ### 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.

23. ### 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. ### What is is neither injective, surjective, and bijective?

As the title says.
25. ### 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. ### 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. ### 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. ### 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. ### 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. ### Injective endomorphism = Surjective endomorphism

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

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. ### 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. ### 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. ### Help with surjective maps

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. ### Norm function is surjective?

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