In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.
A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details.
A function f that is not injective is sometimes called many-to-one.
My attempt:
## ( \rightarrow ) ## Suppose G is injective. Let ## y \in Y ## be arbitrary, denote A = ## \{ y \} ## so that ## G(A) = G(\{ y \}) = f^{-1}[\{ y \}] = \{ x \in X | f(x) \in \{ y \} \} =\{ x \in X | f(x)= y \} ##.
[ However, now I am stuck because I don't know if ## G(A)=...
Definition:
Let ##G## be a graph. ##G## is a functional graph if and only if ##(x_1,y_1) \in G## and ##(x_1,y_2) \in G## implies ##y_1=y_2##.
Problem statement, as written:
Let ##G## be a functional graph. Prove that ##G## is injective if and only if for arbitrary graphs ##J## and ##H##, ##G...
My only qualm is that the statement “Let G be a functional graph” never came into play in my proof, although I believe it to be otherwise consistent. Can someone take a look and let me know if I missed something, please? Or is there another reason to include that piece of information?
I typed this up in Overleaf using MathJax. I'm self-studying so I just want to make sure I'm understanding each concept. For clarification, the notation f^{-1}(x) is referring to the inverse image of the function. I think everything else is pretty straight-forward from how I've written it. Thank...
Hello,
Let f: ]1, +inf[ → ]0, +inf[ be defined by f(x)=x^2 +2x +1.
I am trying to prove f is injective.
Let a,b be in ]1, +inf[ and suppose f(a) = f(b).
Then, a^2 + 2a + 1 = b^2 + 2b + 1.
How do I solve this equation such that I end up with a = b?
Solution:
(a + 1) ^2 = (b + 1)^2...
I'm using the notation T* to indicate the adjoint of T.
I got as far as to say that if T is injective, then T* is surjective. But I don't know how to show that T*T is invertible. Showing that T*T is surjective or injective would imply invertibility, but I'm not sure how to do that either. I...
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...
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'm aware of a typical example of injective immersion that is not a topological embedding: figure 8
##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)##
As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...
I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Chapter 2: Abstract Structures ... ...
I need some help in order to fully understand Awodey Example 2.3, Chapter 2 ... ... Awodey Example 2.3, Chapter 2 reads as follows:
In the Example above Awodey writes the...
Homework Statement
Find, with justification, an injective group homomorphism from ##D_{2n}## into ##S_n##.
Homework EquationsThe Attempt at a Solution
So I'm thinking that the idea is to map ##r## and ##s## to elements in ##S_n## that obey the same relations that r and s satisfy. I can see how...
Let's suppose that I have an element ##e## of order ##p## in the group of complex numbers whose elements all have order ##p^n## for some ##n\in\mathbb{N}## (henceforth called ##K##), and the module generated by ##(e)## is irreducible.
How do I show that the injective hull of the module...
I'm reading a pdf where it's said that the function ##f: \mathbb R \longrightarrow \mathbb{R}^2## given by ##f(x) = \langle \sin (2 \pi x), \cos ( 2 \pi x) \rangle## is not one-to-one, because ##f(x+1) = f(x)##. This is pretty obvious to me. What I don't understand is that next they say that the...
Hey! :o
I want to prove the following criteroin using the mean value theorem for differential calculus in $\mathbb{R}^n$:
Let $G\subset \mathbb{R}^n$ a convex region, $f:G\rightarrow \mathbb{R}^n$ continuously differentiable and it holds that \begin{equation*}\det...
Dear Everybody,
Question:
"Prove that if g(f(x)) is injective then f is injective"
Work:
Proof: Suppose g(f(x)) is injective. Then g(f(x1))=g(f(x2)) for some x1,x2 belongs to C implies that x1=x2. Let y1 and y2 belongs to C. Since g is a function, then y1=y2 implies that g(y1)=g(y2). Suppose...
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...
I have encountered this theorem in Serge Lang's linear algebra:
Theorem 3.1. Let F: V --> W be a linear map whose kernel is {O}, then If v1 , ... ,vn are linearly independent elements of V, then F(v1), ... ,F(vn) are linearly independent elements of W.
In the proof he starts with C1F(v1) +...
Homework Statement
From ##\mathbb{Z}_3## to ##\mathbb{Z}_{15}##
Homework EquationsThe Attempt at a Solution
I know how to do this if we assumed that the rings had to be unital. In that case, there can be no non-trivial homomorphism. However, in my book rings don't need unity, and so a...
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...
Let T be linear transformation from V to W. I know how to prove the result that nullity(T) = 0 if and only if T is an injective linear transformation.
Sketch of proof: If nullity(T) = 0, then ker(T) = {0}. So T(x) = T(y) --> T(x) - T(y) = 0 --> T(x-y) = 0 --> x-y = 0 --> x = y, which shows that...
Homework Statement
Let γ : I → Rn be a regular smooth curve. Show that the map γ is locally injective, that is for all t0 ∈ I there is some ε > 0 so that γ is injective when restricted to (t0 − ε , t0 + ε ) ∩ I.
Homework Equations
The Attempt at a Solution
[/B]
So I know a function (or a...
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...
Hey! :o
Let $F$ be a field and $V,W$ finite-dimensional vector spaces over $F$.
Let $f:V\rightarrow W$ a $F$-linear mapping.
We have to show that $f$ is injective if and only if for each linearly independent subset $S$ of $V$ the Image $f(S)$ is linearly independent in $W$. I have done the...
Hey all, is it possible to find a function that for $$ a,b,c.. \in \mathbb{R} $$ $$ y= f(a,b,c,..) , \hspace{5mm} y= \rho , \rho \in \mathbb{R} \hspace{2mm} for \hspace{2mm} only \hspace{2mm} 1 \hspace{2mm} set \hspace{2mm} of \hspace{2mm} a,b,c.. $$
Any help appreciated
Hi,
I've been trying to find one symmetric "injective" N²->N function, but could not find any. The quotes are there because the function I'm trying to find is not really injective, as I need that the two arguments be interchangeable and the value remains the same.
In other words, the tuple (a...
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.
Homework Statement
Hello,
I need some help on the following. I am BRAND new to set theory and this was in my first HW assignment and I am not sure where to start.
The question is as follows:
Let A and B be parts of a set E
Let P(E)\rightarrow P(A) X P(B) be defined by
f(X)=(A\cap X,B\cap X)...
Homework Statement
##f : A \rightarrow B## if and only if ##\exists g : B \rightarrow A## with the property ##(g \circ f)(a) = a##, for all ##a \in A## (In other words, ##g## is the left inverse of ##f##)
Homework EquationsThe Attempt at a Solution
I have already prove the one direction. Now I...
Homework Statement
Prove that sinx+cosx is not one-one in [0,π/2]
Homework Equations
None
The Attempt at a Solution
Let f(α)=f(β)
Then sinα+cosα=sinβ+cosβ
=> √2sin(α+π/4)=√2sin(β+π/4)
=> α=β
so it has to be one-one
[/B]
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?
Hi! (Wave)
The set $\mathbb{R}$ of real numbers is not countable.
Proof:
We define the function $F: \{0,1\}^{\omega} \to \mathbb{R}$ with the formula:
$$(a_n)_{n \in \omega} \in \{0,1\}^{\omega} \mapsto F((a_n)_{n \in \omega})=\sum_{n=0}^{\infty} \frac{2a_n}{3^{n+1}}$$
Show that $F$ is 1-1...
Homework Statement
The function is ##\phi " \mathbb{Z}_{12} \rightarrow \mathbb{Z}_{24}##, where the rule is ##\phi ([a]_{12}) = [2a]_{24}##. Verify this is a injection
Homework EquationsThe Attempt at a Solution
Let ##[x]_{12} ,[y]_{12} \in \mathbb{Z}_{12}## be arbitrary. Suppose that...
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
Let ## S = \{ (m,n) : m,n \in \mathbb{N} \} \\ ##
a.) Show function ## f: S -> \mathbb{N} ## defined by ## f(m,n) = 2^m 3^n ## is injective
b.) Use part a.) to show cardinality of S.
The Attempt at a Solution
a.) ## f(a,b) = f(c, d ) ; a,b,c,d \in \mathbb{N} \\\\ 2^a...
I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.
I am studying Proposition 28 (D&F pages 387 - 388)
In the latter stages of the proof of Proposition 28 we find the following statement (top of page 388):
"In general, Hom_R (R, X) \cong X...
I am reading Dummit and Foote, Section 10.5 : Exact Sequences - Projective, Injective and Flat Modules.
I am studying Proposition 28 (D&F pages 387 - 388)
In the latter stages of the proof of Proposition 28 we find the following statement (top of page 388):
"In general, Hom_R (R, X) \cong X...
Hello MHB.
I am sorry that I haven't been able to take part in discussions lately because I have been really busy.
I am having trouble with a question.
In a past year paper of an exam I am preparing for it read:
Let $f: (a,b)\to \mathbb R$ be a differentiable function with $f'(x)\neq 0$ for...
Homework Statement
Show that if f: A → B is injective and E is a subset of A, then f −1(f(E) = E
Homework Equations
The Attempt at a Solution
Let x be in E.
This implies that f(x) is in f(E).
Since f is injective, it has an inverse.
Applying the inverse function we see that...
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...
Homework Statement
The function from R to R satisfies x + f(x) = f(f(x)) Find all Solutions of the equation f(f(x)) = 0.
Part of the problem solution says that if f(x) = f(y), then "obviously" x = y. I understand the rest of the solution, but why does f(x) = f(y) imply that x = y?
Homework Statement
The objective was to think of a binary operation ##*:\mathbb{N}\times\mathbb{N}\to\mathbb{N}## that is injective. A classmate came up with the following operation, but had trouble showing it was injective:
##a*b=a^3+b^4##.
Homework Equations
The Attempt at a...
Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. How can i show if the mapping from x to y is injective for given A and alphabet (beside a search method)?
For example, let A and the alphabet be
[1 0 1/sqrt2 1/sqrt2]
[0 1 1/sqrt2 -1/sqrt2]
and...
Homework Statement
Prove or disprove: \exists a binary operation *:\mathbb{N}\times\mathbb{N}\to\mathbb{N} that is injective.
Homework EquationsThe Attempt at a Solution
At first, I was under the impression that I could prove this using the following operation. I define * to be...
Homework Statement
Do all the preimages on X need to have a (and of course I know only one but) image in Y for the f:x->y to be injective?
IS THE FOLLOWING FUNCTION INJECTIVE SINCE ONE ELEMENT OF FIRST DOES NOT HAVE ANY IMAGE
Homework Equations
The Attempt at a Solution
Thank You.