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.
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...
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.
Homework Statement
Find the useful denial of a injective function and a surjective function.
Homework EquationsThe Attempt at a Solution
I know a one to one function is (∀x1,x2 ∈ X)(x1≠x2 ⇒ f(x1) ≠ f(x2)). So would the useful denial be (∃x1,x2 ∈ X)(x1 ≠ x2 ∧ f(x1) = f(x2))?
I know a onto...
wikipedia says:
"The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers R+, then g becomes bijective; its inverse is the...
I have the following question about surjective module-homomorphisms.
Let $f:A \longrightarrow B$ be a surjective $R$-homomorphism between $R$-modules $A$ and $B$.
Let $S, T$ be submodules of $A$ and let $X, Y$ be submodules of $B$.
I can prove that in general
$$f(S+T)=f(S)+f(T)$$
and in...
I am reading Reinhold Remmert's book "Theory of Complex Functions" ...
I am focused on Chapter 0: Complex Numbers and Continuous Functions ... and in particular on Section 1.4: Angle-Preserving Mappings ... ...
I need help in order to fully understand a remark of Remmert's regarding...
Hi, I found in Kreyszig that if for any ##x_1\ and\ x_2\ \in \mathscr{D}(T)##
then an injective operator gives:
##x_1 \ne x_2 \rightarrow Tx_1 \ne Tx_2 ##
and
##x_1 = x_2 \rightarrow Tx_1 = Tx_2 ##If one has an operator T, is there an inequality or equality one can deduce from this, in...
Homework Statement
I have attached the question. Translated: Suppose T: R^4 -> R^4 is the image so that: ...
Homework Equations
So I did this question and my final answers were correct: 1. not surjective 2. not injective. My method of solving this question is completely different than the...
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...
Stumped on a couple of questions, if anyone could help!
In what follows I will denote the identity function; i.e. I(x) = x for all x ∈ R.
(a) Show that a function f is surjective if and only if there exists a function g such that f ◦ g = I.
(b) Show that a function f is injective if and only if...
Homework Statement
Prove that an endomorphism between two finite sets is injective iff it is surjective
Homework EquationsThe Attempt at a Solution
I can explain this in words. First assume that it is injective. This means that every element in the domain is mapped to a single, unique element...
Hello guys! I'm taking Discrete Mathematics this semester and I got this question in one of my homework tasks.
I've tried thinking about the solution over and over but can't seem to come up with anything..
The question goes like this: Is there a Surjective function from [0,1]\{1/2} to [0,1] such...
Hey! :o
I am looking at the following exercise:
Let $C$ be an algebraic closure of $F$, let $f\in F[x]$ be irreducible and let $a,b\in C$ be roots of $f$.
Applying the theorem:
"If $E$ is an algebraic extension of $F$, $C$ is an algebraic closure of $F$, and $i$ is an embedding (that is, a...
I have a mapping $L: \mathbb R^3 \rightarrow \mathbb R^3$ as defined by $L(x, y, z) = (x+z, y+z, x+y).$ How do you prove that the $L$ is an onto mapping? I know for sure that $\forall x, y, z \in \mathbb R$, then $x+z, y+z, x+y \in \mathbb R$ too. Then I need to prove that $Im (L) = \mathbb...
Homework Statement
Count the number of surjective functions from {1,2,...,n} to {a,b,c,d}. Use a formula derived from the following four-set venn diagram:
Homework Equations
None provided.
The Attempt at a Solution
First, I divided the Venn diagram into sets A,B,C,D and tried to express...
Homework Statement
Show that if ##f:A\rightarrow B## is subjective and ##H\subseteq B## then ##f(f^{-1}(H))=H##, give an example to show the equality need to hold if ##f## if not surjective.
Homework Equations
3. The Attempt at a Solution [/B]
I know that I want to show that an element...
Hello all,
Can anyone give me a pointer on how to start this proof?:
f:E\rightarrow F we consider f^{-1} as a function from P(F) to P(E).
Show f^(-1) is injective iff f is surjective.
Hello,
I wonder if anyone could settle a disagreement I'm having with one of my peers. The question is 'How many surjective functions are there from a set of size n+3 to a set of size n?'. Now, I've already proven that there are (n+1 choose 2)n! surjective functions from a set of size n+1 to a...
Hello,
I've been reading about injectivity from Z to N and surjectivity from N to Z and was wondering whether there was some kind of algorithm that could generate these specific types of functions?
Homework Statement
$$f:\mathbb{R}\rightarrow\mathbb{R}~~\text{where}~~f(x)=x^3+2x^2-x+1$$
Show if f is injective, surjective or bijective.
Homework EquationsThe Attempt at a Solution
f is obviously not injective (and thus not bijective), one counter example is x=-1 and x=1.
I can see from the...
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...
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##...
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...
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...
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...
Homework Statement
Let PosZ = {z ∈ Z : z > 0}. Consider the function f : PosZ → PosZ dened 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...
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...
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...
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...
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...
[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) =...
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...
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...
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...
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...
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...
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...
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.
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}...
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...
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...
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...
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?
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...