Injective Definition and 97 Threads

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...

is the function x³-5x²+3x+5 injective. how can you tell

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...

As the title says.

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...

Is an injective endomorphism necessarily surjective? And it is also true the opposite?

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...

I quote an unsolved question from MHF posted by user jackGee on February 3rd, 2013. P.S. Of course, I meant in the title and instead of an.

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.