Injective Definition and 97 Threads

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.

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  1. D

    Example of a linear transformation L which is injective but not surj, or vice versa

    Homework Statement Give an example of a linear vector space V and a linear transformation L: V-> V that is 1.injective, but not surjective (or 2. vice versa) Homework Equations -If L:V-> V is a linear transformation of a finitedimensional vector space, then L is surjective, L is...
  2. J

    A homomorphism is injective if and only if its kernel is trivial.

    I was a little curious on if I did the converse of this biconditonal statement correctly. Thanks in advance! =)Proposition: Suppose f:G->H is a homomorphism. Then, f is injective if and only if K={e}. Proof: Conversely, suppose K={e}, and suppose f(g)=f(g’). Now, if f(g)=f(g’)=e, then it follows...
  3. G

    Can Wolfram Rule 90 be Made Reversible with a Simple Modification?

    http://en.wikipedia.org/wiki/Rule_90 Wolfram Rule 90 is a type of cellular automata. Each cell's value is computed as the XOR of its two neighbors in the predecessors generation. Rule 90 cannot be reversed, because a given configuration has 4 possible predecessor configurations. However, I've...
  4. T

    Injective Function Proof for Decreasing Functions

    Homework Statement Let E ⊆ R and f : E → R a decreasing function for all x ∈ E. Prove that f is injective. The attempt at a solution I tried that f were not injective. Then, there exist x < y such that f(x) = f(y) -This contradicts f being a decreasing function. I think this is...
  5. H

    An injective function going from N to the set of algebraic numbers

    Homework Statement Prove that the set of algebraic numbers is countably infinite. Homework Equations If there exists a bijective map between N and a set A, N and A have the same cardinality The Attempt at a Solution Rather than coming up with a bijective map between S =the set of...
  6. D

    Proving Injection in Composite Functions

    Given two functions: f:A --> B g:B --> C How to show that if the (g ° f) is injection, then f is injection? I tried this: We need to show that g(f(a)) = g(f(b)) ==> a = b holds true for all a, b in A. But there's nothing said about function g.
  7. K

    Injective and Continuity of split functions

    Homework Statement Let I:=[0,1], let f: I→ℝ defined by f(x):= x when x is rational and 1-x when x is irrational. Show that f is injective on I and that f(fx) =x for all x in I. Show that f is continuous only at the point x =1/2 **I think i addressed all of these questions but I am unsure...
  8. L

    MHB Proving F is Not Injective in R^n

    Hello. I have the next problem: Let $f: U\subset R^n\rightarrow R$ a class $C^1$ function in an open subset $U$ of $R^n$. Proff that f can't be injective.There are some idications: suppose that the vector ($\nabla f$)(p) is not zero (if it's zero the function is not injective WHY?) in $p\in U$...
  9. phoenixthoth

    Injective homomorphism into an amalgam of structrues

    Hello all This question relates to products of structures all with the same symbol set S. After I give a little background the question follows. *Direct Products* This definition of the direct product is taken from Ebbinghaus, et.al. Let I be a nonempty set. For every i\in I, let...
  10. P

    Show that linear transformation is surjective but not injective

    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.
  11. A

    Injective and Surjective linear transformations

    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}...
  12. P

    Continuous Injective Function on Compact Set of C

    Prove that the inverse of a continuous injective function f:A -> ℂ on a compact domain A ⊂ ℂ is also continuous. So basically because we're in ℂ, A is closed and bounded, and since f is continuous, the range of f is also bounded. Given a z ∈ A, I can pick some arbitrary δ>0 and because f is...
  13. P

    F is diffeomorphism implys df is injective?

    Let U be a non-empty open set in Rn, if f:U->Rm is a diffeomorphism onto its image, show that df(p) is injective for all p in U. How can I attack this problem?
  14. G

    Composition of Mappings, Surjective and Injective

    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...
  15. E

    "Intersection Equality iff Function is Injective

    Homework Statement Let A, B be sets, C,D\subset A and f:A\longrightarrow B be a function between them. Then f(C\cap D)=f(C)\cap f(D) if and only if f is injective. Homework Equations Another proposition, that I have proven that for any function f(C\cap D)\subset f(C)\cap f(D), and the...
  16. T

    Why do irrational numbers have unique decimal expansions?

    Hello, I am looking for an example of two input injective function, f(x1,x2), R x R ->R. I am very grateful if you can find one. Thanks
  17. G

    Contour line of Injective function

    Hello, I have g(t) is a continuous and a differential function under 1 variable. let h(x,y)=g(x^2+y^2) suppose that g(t) is Injective (thus monotonous) What is the shape of the contour lines of the graph of h(x,y)? -I have a sense that we're talking about simple cycles but I...
  18. C

    Why is the Injective Operator in L^2(0,1) One-to-One?

    Hi! Define an integral operator K: L^2 (0,1) -> L^2(0,1) by: Kx(t) = Integral[ (1+ts)exp(ts)x(s) ds from t=0 to t=1]. Why is "obvious" that K is a one-to-one operator? I know K is one to one if Kx(t) = 0 implies x(t) = 0 but I don't see why this is true. Can you please explain why?
  19. M

    How Can Injectivity Prove f^(-1)(f(A)) = A?

    who can help me? ı want to prove this If f : X → Y is injective and A is a subset of X, then f −1(f(A)) = A. but how can I do this :(
  20. H

    Field of modulo p equiv classes, how injective linear map -> surjectivity

    Field of modulo p equiv classes, how injective linear map --> surjectivity Homework Statement Let Fp be the field of modulo p equivalence classes on Z. Recall that |Fp| = p. Let L: Fpn-->Fpn be a linear map. Prove that L is injective if and only if L is surjective. Homework Equations...
  21. M

    How can we prove the injectivity of f(x) = x³ + x without using calculus?

    Hi, How do I prove that this functions is injective? a.) f : x --> x³ + x x ∈ R f(a) = a³ + a, f(b) = b³ + b f(a) = f(b) => a³ + a = b³ + b => a³ = b³ => a = b therefore f is one-to-one
  22. X

    a) Falseb) False c) True d) True

    Homework Statement Mark as true or false. (a) A function is injective if a 6\neq b yields f(a) 6\neq f(b). (b) A function is injective if f(a) = f(b) in case that a = b. (c) A function is injective if f(a) = f(b) only if a = b. (d) A function is injective only when f(a) 6\neq f(b) yields a...
  23. I

    If a matrix A is injective then AAt is invertible

    Homework Statement If a matrix, A (nxm) is monic (or epic) then is A^tA (or AA^t) is invertible? Homework Equations T is monic if for any matrices B,C: BT = BT => B=C. S is invertible if there exists U s.t. US = SU = I_n The Attempt at a Solution Since A is monic it must...
  24. S

    Proving Linear Injectivity in Finite-Dimensional Vector Spaces

    Hey guys, new to the forum but hoping you can help. How do you prove that vector spaces V and U have a linear injective map given V is finite dimensional. I got the linear part but cannot really figure out the injectivity part, although I am thinking that it has to do with the kernel...
  25. X

    Is this function injective, surjective, or both?

    Homework Statement The following function f is a function from R to R. Determine whether f is injective (one-to-one), surjective (onto), or both. Please give reasons. Homework Equations f(x) = (x+1)/(x+2) if x != -2 f(x) = 1 when x = 2 The Attempt at a Solution f'(x) = 1/(x+2)2...
  26. G

    G of f injective, but g not injective

    Homework Statement Give an example of a map f:A\rightarrowB and a map g:B\rightarrowC where g of f is injective but g is not injective. Homework Equations The Attempt at a Solution I'm not really sure what they are asking for.
  27. B

    Injective linear transformation

    Homework Statement We regard each polynomial p(t) an element of R(t) as defining a function p:R\rightarrow R, x \rightarrow p(x) prove that g:R[t]\rightarrow R[t], p(t) \rightarrow \int_{0}^{t}p(x)dx defines an injective linear transformation. Homework Equations The...
  28. E

    Prove that T is injective if and only if T* is surjective

    Homework Statement T ∈ L(V,W). Thread title. Homework Equations The Attempt at a Solution Note that T* is the adjoint operator. But there's one thing that I need to get out of the way before I even start the proof. Now consider <Tv, w>=<v, T*w> w in W, v in V. Now when they say T...
  29. M

    How Do You Determine if a Linear Transformation is Injective?

    I was just wondering how you know if linear transformations injective?
  30. F

    Am I right in my injective and surjective definition?

    In layman terms otherwise I have trouble understanding Injective: A function where no element on the domain is many to one. Surjective: All the elements in the codomain have at least one element from the domain that maps to them. I'd like to keep it simple so I can play it back to...
  31. R

    Three questions on injective functions.

    Q1. Claim: Suppose f : Rn -> Rm is injective. Then m >= n. Is this true? Q2. Claim: Suppose f : Rn -> Rn is injective and f(X) = [f1(X) f2(X) ... fn(X)]T. Then each fk must be injective. Is this true? Q3. I assume the above claims are known results or have known counterexamples. Can...
  32. T

    What is the relationship between f-1(f(A0)) and A0 in terms of injectivity?

    I'm not sure how i would go about this problem... Let f: A-> B (which i know means... f is a function from A to B which also means... that A is the domain and B is the range or image) Let A0\subsetA and B0\subsetB a. show that A0\subsetf-1(f(A0)) and the equality hold if f is...
  33. R

    Surjective, injective and predicate

    Homework Statement How do I check if my function is surjective? How do I check if my function is injective? Suppose my function is a predicate and hence characteristic function of some set. How do I determine such a set? Homework Equations Does anyone know to write "The function...
  34. J

    Proof that a specific map is an injective immersion

    Homework Statement Consider f: R^{m+1} - {0} -> R^{(m+1)(m+2)/2}, (x^{0},...,x^{m}) -> (x^{i} x^{j}) i<j in lexicografical order a) prove that f is an immersion b) prove that f(a) = f(b) if and only if b=±a, so that f restricted to Sm factors through an injective map g from Pm. c) show g...
  35. C

    Ker(phi) = {0}, then phi injective?

    Hi all, Can anyone point to an explanation of why if ker(phi) = {0}, then phi is injective?
  36. G

    The module is injective iff it is a direct summand of an injective cogenerator

    could anyone give me a proof of this statement: The module is injective iff it is a direct summand of an injective cogenerator.
  37. quasar987

    Proving Injectivity of the Map T: L^p(E) --> (L^q(E))* for 1<p<2 and q>=2

    [SOLVED] Show map is injective Homework Statement Going crazy over this. Let 1<p<2 and q>=2 be its conjugate exponent. I want to show that the map T: L^p(E) --> (L^q(E))*: x-->T(x) where <T(x),y> = \int_Ex(t)y(t)dt is injective. This amount to showing that if \int_Ex(t)y(t)dt=0 for all...
  38. Y

    Can u gave me some examples of injective function.

    can u gave me some examples of injective function that is not surjective. Is f(x)=y a injective function that is not surjective?
  39. Y

    What are injective and surjective maps in vector spaces?

    Hello! I hope I've posted this in the correct section... I'm a 3rd year undergraduate and we're currently studying Vector Spaces (in QM) and I just don't understand what injective (one-to-one) and surjective (onto) mean? As a result I have no idea what an isomorphism is! I realize this is...
  40. C

    I'm trying to prove that a linear map is injective

    hello, I've been reading some proofs and in keep finding this same argument tyo prove that a linear map is injective viz, we suppose that t(a,c) = 0 and then we deduce that a,c = 0,0. is it the case that the only way a linear map could be non injective is if it took two elements to zero? i.e. t...
  41. W

    How can the inequality hold for an injective function?

    This is something I understood before, but for some reason I forgot it. How do you prove this inequality holds, if f is injective? A_0 \subset f^{-1}(f(A_0))
  42. I

    Prove Injectivity of x^x Function?

    I have recently taken great interest in studying the properties of the function f(x) = x^x , and I was wondering: is there any way to prove whether f(x) = x^x is an injective (i.e. one-to-one) function? I realize that if I can prove that if the inverse of f(x) = x^x is also a function, then...
  43. H

    Surjective, injective, bijective how to tell apart

    Hi, I have no problems with recognising a bijective function -> one-to-one mapping e.g. x^3 is bijective wheras x^2 is not. But how do you tell weather a function is injective or surjective? I was reading various "math" stuff on this but it has left me only puzzled. Can somebody explain...
  44. G

    Linear Transformation: R^n to R^m - Injective?

    Indicate whether each statement is always true, sometimes true, or always false. IF T: R^n --> R^m is a linear transformation and m > n, then T is 1-1 Not sure to how prove this..
  45. K

    Proof of Injective Function Property

    I had this question on a test today. Prove that if a function f:X-->Y is injective, then f(X\setminus A) \subset Y\setminus f(A), \forall A \subset X. This is how I did it: If x_1 is in A, then y_1=f(x_1) is in f(A). Because the function is injective, we can pick (cut Y into pieces) f(A) and...
  46. J

    Injective Function: Cubic Function Real Numbers?

    is a cubic function injective for all real numbers?
  47. L

    Bijections result when the function is surjective and injective

    Bijections result when the function is surjective and injective. How do I find a bijection in N and the set of all odd numbers? f(x) = 2x+1 Do I have to prove that this is one-to-one and onto? Am I on the right track?
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