Surjection Definition and 26 Threads

a) Yes. One surjection from ##\mathbb{Z}## to ##\mathbb{N}## is the double cover of ##\mathbb{N}## induced by ##f:\mathbb{Z}\longmapsto\mathbb{N}## with $$f(z)=\begin{cases} -z & ,\forall z<0\\ z+1 & ,\forall 0\leq z \end{cases}$$ b) Yes. One surjection from ##\mathbb{R}## to ##\mathbb{N}## is...

1) Two sets have the same cardinality if there exists a bijection (one to one correspondence) from ##X## to ##Y##. Bijections are both injective and surjective. Such sets are said to be equipotent, or equinumerous. (credit to wiki) 2) ##|A|\leq |B|## means that there is an injective function...

In my book, the definition of surjection is given as follows: Let A and B be sets and f:A->B. The function f is said to be onto if, for each b ϵB, there is at least one a ϵ A for which f(a)=b. In other words, f is onto if R(f)=B. A function which is onto is also called a surjection or a...

If f were a function of 1 variable only, then this would be straight forward as I can try to find its inverse by reversing the operations defined in f. I know I need to show that for any given positive integer,p, there exists two positive integers, m and n such that 1/2(m+n−2)(m+n−1)+n=p...

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...

Homework Statement Proof that: f has an inverse ##\iff## f is a bijection Homework Equations /definitions[/B] A) ##f: X \rightarrow Y## If there is a function ##g: Y \rightarrow X## for which ##f \circ g = f(g(x)) = i_Y## and ##g \circ f = g(f(x)) = i_X##, then ##g## is the inverse...

This is a rather simple question, so it has been rattling my brain recently. Consider a surjective map ## f : S \rightarrow T ## where both ## S ## and ## T ## are finite sets of equal cardinality. Then is ## f ## necessarily injective? I proved the converse, which turned out to be quite...

Homework Statement Let be linear surjection. Prove that then n>=m. Homework Equations Definition(surjection): The Attempt at a Solution Lets assume opposite, n<=m. If that is the case, then for some y from R^m, there is no belonging x from R^n, what is in contradiction with definition where...

Homework Statement Let A and B be two sets. Homework Equations Prove that there exists a injection from A to B if and only if there exists a surjection from B to A The Attempt at a Solution I did one implication which is we suppose that f: B→A is a surjection. Then by definition of a...

Can anyone explain to me how to do these types of questions? I have the answers but I don't understand it. The function f: N -> N, f(n) = n+1 is (a) Surjection but not an injection (B) Injection but not a surjection (c) A Bijection (d) Neither surjection not injection The answer is B...

Hi, I was wondering whether the following is true at all. The first isomorphism theorem gives us a relation between a group, the kernel, and image of a homomorphism acting on the group. Could this possibly also imply that there exists a surjective homomorphism either mapping the previous kernel...

Homework Statement How can I prove this? If g°f is a bijective function, then g is surjective and f is injective. Homework Equations The Attempt at a Solution

Hi, Everyone: I am reading a paper that refers to a "natural surjection" between M_g and the group of symplectic 2gx2g-matrices. All I know is this map is related to some action of M_g on H₁(S_g,Z). I think this action is...

Surjection from N --> Z Homework Statement Find a surjection map, f: N -> Z Homework Equations The Attempt at a Solution I think this is equivalent to finding an injection map: g: Z -> N So I defined it: g(z) = -z, if z is negative z, if z is positive Is this...

how do you prove injection and surjection of the function of 2 variables. for example f:RxR->R

Homework Statement Are the given functions injective? Surjective? a) seq: N -> Lists[N] b) f: Lists[A] -> P(A), f(x)=(<x1,x2,...,xn>)={x1,x2,...,xn} Homework Equations The Attempt at a Solution a) Ok so the domain contains a sequence of natural numbers. and the range contains a list? What...

I'm having trouble understanding just what is the difference between the three types of maps: injective, surjective, and bijective maps. I understand it has something to do with the values, for example if we have T(x): X -> Y, that the values in X are all in Y or that some of them are in Y...

Hi :smile: I'm new on these forums, and not only is this my first post, but this is also my first thread. The following is not a homework question, but a question I found. However, I have no idea how to do this. I would appreciate it if someone could help me. Please click on the following...

Hi, there's a proposition in my lecture notes which states "If X is a set, there can be no surjection X \rightarrow P(X)", where P(x) is the power set of X (does anyone know how I get the squiggly P?). The proof given there seems unnecessarily complicated to me. Would the following be...

f(x) is a bijection if and only if f(x) is both a surjection and a bijection. Now a surjection is when every element of B has at least one mapping, and an injection is when all of the elements have a unique mapping from A, and therefore a bijection is a one-to-one mapping. Let's say that...

Homework Statement Find a continuous surjection from [0,1) onto [0, infinity) Homework Equations The Attempt at a Solution I have only been able to come up with one mapping but then I realized it did not work. Any help would be appreciated.

Homework Statement Suppose that f is a mapping from a finite set X to P(X) (the power set of X). Prove that f is not surjective.Homework Equations The Attempt at a Solution My proof strategy is 1) Show that P(X) always has more elements than X 2) Show that a mapping from X->Y cannot be...

If a function f: N-->X is surjective , is f^-1(X) (its inverse image) also surjective? If so, why?

I was just looking at the definitions of these words, and it reminded me of some things from linear algebra. I was just wondering: Is a bijection the same as an isomorphism?

I have a linear map from $ V\rightarrow K[X_{1},...,X_{n}]\rightarrow K[X_{1},...,X_{n}]/I.$ how do i prove that a linear map from $ V=\{$polynomials with $\deg _{x_{i}}f\prec q\}$ to $ K[X_{1},..X_{n}]/I.$ where I is the ideal generated by the elements $ X_{i}^{q}-X_{i},1\leq i\leq n.,$ is...

Ring, field, injection, surjection, bijection, jet, bundle. Does anybody know who first introduced those terms and when and why those people called these matimatical structures so. I mean not the definitions but the properties of real things which can be accosiated with those mathematical...