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. S

    Solving Surjective Functions Homework

    Homework Statement i) Show explicitly that any non-injective function with a right inverse has another right inverse ii) Give an example of a function which has (at least) two distinct left inverses Homework Equations i) I don't believe there are any. ii) " The Attempt at a Solution...
  2. S

    Confusion about defn. of Surjective mapping in WIKI.

    Reference: http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection Consider the two sets X & Y connected by a the relation y^2=x^2. (For simplicity we can take X={-2,2} and Y={-2,2}).Then can we call the mapping from X to Y to be surjective? From the definition of WIKI, the answer...
  3. A

    A Surjective Function for All [R]

    Hi, I wonder if there are any surjective functions whose range covers all real numbers. Note that I'm not implying countability, since |R| is uncountable. For instance, Sum[rn/n!, {n, 0, m}] : {r, n, m} ∈ R Thanks for help...
  4. radou

    Closed continuous surjective map and Hausdorff space

    Homework Statement Here's a nice one. I hope it's correct. Let p : X --> Y be a closed, continuous and surjective map such that p^-1({y}) is compact for every y in Y. If X is Hausdorff, so is Y. The Attempt at a Solution Let y1 and y2 in Y. p^-1({y1}) are then p^-1({y2}) disjoint...
  5. radou

    Closed continuous surjective map and normal spaces

    Homework Statement Let p : X --> Y be a closed, continuous and surjective map. Show that if X is normal, so is Y. The Attempt at a Solution I used the following lemma: X is normal iff given a closed set A and open set U containing A, there is an open set V containing A and whose...
  6. atomqwerty

    Surjectivity of a Three-Dimensional Function with Non-Negative Real Inputs

    Hello, is this function surjective? \Phi : \Re^{+} \diamond \Re \diamond \Re \rightarrow \Re^{3} (r,\varphi,\theta) \rightarrow (r cos\varphi sin\theta, r sin\varphi sin\theta, r cos\theta) PS Diamond means X (cross)
  7. L

    Solving for a Surjective Matrix

    I saw this in a book as a Proposition but I think it's an error: Assume that the (n-by-k) matrix, A, is surjective as a mapping, A:\mathbb{R}^{k}\rightarrow \mathbb{R}^{n}. For any y \in \mathbb{R}^{n} , consider the optimization problem min_{x \in \mathbb{R}^{k}}\left{||x||^2\right} such...
  8. L

    Solving for Surjective Matrix: A Possible Typo in the Theorem Statement?

    I saw this in a book as a proposition but I think it's an error: Assume that the (n-by-k) matrix, A, is surjective as a mapping, A:R^{k}\rightarrow R^{n}. For any y \in R^{n} , consider the optimization problem min_{x \in R^{k}}\left{||x||^2\right} such that Ax = y. Then, the following...
  9. A

    Cancellation law with surjective functions

    Homework Statement suppose function f : A \to B, g: A \to B, h : B \to C satisfy g \circ f=h \circ f. If is surjective then prove that g=h Homework Equations n/a The Attempt at a Solution so for any x \in A, gf(x)=hf(x), and for any b \in B there exist a \in A, such that f(b)=a...
  10. E

    If g o f is surjective, then is f surjective?

    Homework Statement Assume f:A\rightarrowB g:B\rightarrowC h=g(f(a))=c Give a counterexample to the following statement. If h is surjective, then f is surjective. Homework Equations Definition ofSurjection: Assume f:A\rightarrowB, For all b in B there is an a in A such that f(a)=b...
  11. D

    I need to prove that the following is not surjective. how do i do

    i need to prove that the following is not surjective. how do i do that? let f:R->R be the function defined by f(x)=x^2 + 3x +4.
  12. J

    Surjective Functions: Understanding Domain and Range

    Consider the function f: Z -> Z, where f(x) 4x+1 for each x is an element in Z, here the range of F = { ... -8, -5, -2, 1, 4, 7...} is a proper subset of Z, so f is not an onto (surjective) function. When one examines 3x + 1 = 8, we know x = 7/3, so there is no x in the domain Z with f(x) =...
  13. R

    Ratio of functions, surjective (analysis course)

    Homework Statement let f: R->R be a continuous function Suppose k>=1 is an integer such that lim f(x)/x^k = lim f(x)/x^k = 0 x->inf x->-inf set g(x)= x^k + f(x) g: R->R Prove that (i) if k is odd, then g is surjective (ii) if k is even, then there is...
  14. B

    Continuous expansion is surjective

    Homework Statement (X,d) a compact metric space, f:X->X cts fn, with d(f(x),f(y)) >= d(x,y) for all x, y in X Prove that f is a surjection.The Attempt at a Solution Let K be the set of points that are not in f(X). It is a union of open balls because X is closed and hence so is f(X). Choose...
  15. X

    Is this function injective, surjective, or both?

    Homework Statement The following function f is a function from R to R. Determine whether f is injective (one-to-one), surjective (onto), or both. Please give reasons. Homework Equations f(x) = (x+1)/(x+2) if x != -2 f(x) = 1 when x = 2 The Attempt at a Solution f'(x) = 1/(x+2)2...
  16. J

    Quick question about surjective functions

    Question Details: f(a/b) = 2^a * 3^b where (a/b) is in lowest terms. Show f is surjective (onto). Note: f maps positive integers to natural numbers --- Is it sufficient to say that... It is onto because for every natural number y there is (a/b) s.t. f(x) = y.
  17. E

    Prove that T is injective if and only if T* is surjective

    Homework Statement T ∈ L(V,W). Thread title. Homework Equations The Attempt at a Solution Note that T* is the adjoint operator. But there's one thing that I need to get out of the way before I even start the proof. Now consider <Tv, w>=<v, T*w> w in W, v in V. Now when they say T...
  18. S

    Basis for the image of a surjective linear map.

    Homework Statement Let V and W be vector spaces over F, and let T: V -> W be a surjective (onto) linear map. Suppose that {v1, ..., v_m, u1, ... , u_n} is a basis for V such that ker(T) = span({u1, ... , u_n}). Show that {T(v1), ... , T(v_m)} is a basis for W. Homework Equations Basic...
  19. A

    Example of dense non surjective operator

    Hi, can anyone give me an example of a bounded operator on a Hilbert space that has dense range but is not surjective? (Preferably on a separable Hilbert space) Im pretty sure such an operator exists since the open mapping theorem requires surjectivity and not just dense range, but its just...
  20. F

    Am I right in my injective and surjective definition?

    In layman terms otherwise I have trouble understanding Injective: A function where no element on the domain is many to one. Surjective: All the elements in the codomain have at least one element from the domain that maps to them. I'd like to keep it simple so I can play it back to...
  21. D

    Surjective Function: A to B Mapping

    For a sirjective function from A--> B, I was just wondering if more than one elements in B can point to the same element in A if the function is surjective.
  22. T

    What is the relationship between f-1(f(A0)) and A0 in terms of injectivity?

    I'm not sure how i would go about this problem... Let f: A-> B (which i know means... f is a function from A to B which also means... that A is the domain and B is the range or image) Let A0\subsetA and B0\subsetB a. show that A0\subsetf-1(f(A0)) and the equality hold if f is...
  23. M

    Prove Surjective function (R:reals) with |f(x)-f(y)|>k|x-y|

    (R:reals) Let f:R-->R be continuous and satisfy |f(x)-f(y)|>or eq. to k|x-y| for all x, y in R and some k>0. Show that f is surjective. I can show that f is injective: let f(x) = f(y), hence k|x-y|< or eq. to 0, thus x=y. I had a suggestion that it might be helpful to show that f has...
  24. R

    Surjective, injective and predicate

    Homework Statement How do I check if my function is surjective? How do I check if my function is injective? Suppose my function is a predicate and hence characteristic function of some set. How do I determine such a set? Homework Equations Does anyone know to write "The function...
  25. A

    Is this a surjective homomorphism?

    I'm trying to prove that if M,N are normal in G and MN = G, then G/(M\cap N)\cong G/M \times G/N In an attempt to use the 1st Isom. Thm, I have a homomorphism from G to G/M x G/N : g \mapsto (gM, gN) The kernel is M\cap N, so I just have to show that the function is onto to get the...
  26. nicksauce

    Prove Surjectivity of g∘f: Homework Solution

    Homework Statement Let f:X\rightarrow~Y and g:Y\rightarrow~Z be surjections. Show that g\circ~f is surjective. Homework Equations The Attempt at a Solution Proof: Suppose f and g are surjections. Then (1)\forall~y\in~Y \exists~x\in~X\textnormal{ st. }f(x)=y And (2)...
  27. T

    Cartesian product & Surjective functions

    I'm a bit stuck here, my question asks me to prove that the product of 2 enumerable sets is indeed enumerable with an argument or a counterexample. I pretty much have no idea on how to proceed, although i know that the product is enumerable
  28. Y

    What are injective and surjective maps in vector spaces?

    Hello! I hope I've posted this in the correct section... I'm a 3rd year undergraduate and we're currently studying Vector Spaces (in QM) and I just don't understand what injective (one-to-one) and surjective (onto) mean? As a result I have no idea what an isomorphism is! I realize this is...
  29. Z

    Surjective Homomorphisms of Coordinate Rings

    Homework Statement I want to show that the homomorphism phi:A(X)->k+k given by taking f(x_1,...,x_n)-> (f(P_1),f(P_2)) is surjective. That is, given any (a,b) in k^2 (with addition and multiplication componentwise) I want to find a polynomial that has the property that f(P_1)=a and f(P_2)=b...
  30. E

    Surjective group homomorphism

    Homework Statement Show the map (call it phi) from U_n to C* defined by phi(X) = det(X) for all matrices X in U_n, is a surjective homomorphism, where U_n is the subgroup of GL(n,C) consisting of unitary matrices C* = C\{0} = invertible/nonzero complex numbers det(.) is the...
  31. S

    How do you prove that a function is surjective?

    how do you prove that a function is surjective ? i know that surjective means it is an onto function, and (i think) surjective functions have an equal range and codomain?
  32. H

    Surjective, injective, bijective how to tell apart

    Hi, I have no problems with recognising a bijective function -> one-to-one mapping e.g. x^3 is bijective wheras x^2 is not. But how do you tell weather a function is injective or surjective? I was reading various "math" stuff on this but it has left me only puzzled. Can somebody explain...
  33. S

    [Discrete Math] f: A->B; surjective? find necessary & sufficient condition.

    [Discrete Math] f: A-->B; surjective? find necessary & sufficient condition. Ok in practice for my discrete exam, I have the following problem. Let f : A->B be a function. a) Show that if f is surjective, then whenever g o f = h o f holds for the functions g,h : B -> C, then g =h. b)...
  34. L

    Bijections result when the function is surjective and injective

    Bijections result when the function is surjective and injective. How do I find a bijection in N and the set of all odd numbers? f(x) = 2x+1 Do I have to prove that this is one-to-one and onto? Am I on the right track?
  35. B

    Surjective and bijective mapping

    Hi, can anyone tell me what a surjective mapping between Hilbertspaces is? That is: what does surjective mean? What about bijective? I mean what is special about a mapping if it is sujective or bijective?
  36. quasar987

    How to show tanx is surjective?

    How can I show the function f:]-\frac{\pi}{2},\frac{\pi}{2}[ \rightarrow R defined by f(x)=tan(x) is surjective? If the domain was a closed interval I could use the intermediate value theorem, but now? Thank you.
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