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.
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<sub>g</sub>
and the group of symplectic 2gx2g-matrices. All I know is this map is related to some
action of M<sub>g</sub> on H<sub>1</sub>(S<sub>g</sub>,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...
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...
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...