Surjective Definition and 83 Threads

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...

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...

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)

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...

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...

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...

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...

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.

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) =...

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...

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...

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...

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.

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...

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...

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...

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...

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.

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...

(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...

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...

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...

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)...

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

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...

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...

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...

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?

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...

[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)...

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?

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?

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.