What is Bijection: Definition and 99 Discussions

In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. The term one-to-one correspondence must not be confused with one-to-one function (an injective function; see figures).

A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets.
A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms the symmetric group.
Bijective functions are essential to many areas of mathematics including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group, and projective map.

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  1. G

    How can I prove it? (injection, bijection, surjection)

    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
  2. B

    Continuous Bijection f:X->X not a Homeo.

    Continuous Bijection f:X-->X not a Homeo. Hi, All: A standard example of a continuous bijection that is not a homeomorphism is the map f:[0,1)-->S^1 : x-->(cosx,sinx) ; for one, S^1 is compact, but [0,1) is not,so they cannot be homeomorphic to each other. Now, I wonder...
  3. S

    Bijection for a power set function

    Homework Statement Let N={1,2...n} .Define the Power set of N,P(N). a) show that the map f:P(N)->P(N) defined by taking A to belong to P(N) to N\A is a bijection. b)C(n,k)=C(n,n-k). The Attempt at a Solution Now the power set is defined by P(N)=2^n and a bijection is a one-to-one...
  4. A

    Mapping a general curve onto a bijection.

    Homework Statement Denote the xy plane by P. Let C be some general curve in P defined by the equation f ( x , y ) = 0 where f ( x , y ) is some algebraic expression involving x and y. Verify carefully that if B : P -> P is any bijection then B( C ) is defined by the equation f ( B^-1...
  5. P

    A nonempty set T1 is finite if and only if there is a bijection from T1->T2?

    Homework Statement Prove that a nonempty set T1 is finite if and only if there is a bijection from T1 onto a finite set T2. The Attempt at a Solution There are at least two different solutions to this problem that I found online...
  6. B

    Complex Analytic Bijection: Is it a Local Diffeo?

    Hi, Everyone: Say f(z) defined on a region R , is a complex-analytic bijection. Does it follow that f:R--->f(R) is a diffeomorphism, i.e., is f<sup>-1</sup> also analytic? I know this is not true for the real-analytic case, e.g., f(x)=x<sup>3</sup> , but complex- analytic...
  7. I

    Proving T is a Bijection: D* to D

    Homework Statement D* = {(u,v) | u>0, v>0, u + v < pi/2} and D = {(x,y) | 0<x<1, 0<y<1} The map T : R2 -> R2 is given by (x,y) = T(u,v) = (sin u/cos v, sin v/cos u). To prove that T is a bijectionHomework Equations and The Attempt at a Solution I'm kinda stuck finding the limits for u and v...
  8. N

    Proving that an equivalence relation is a bijection

    Homework Statement Let (a, b), (c, d) be in R x R. We define (a, b) ~ (c, d) iff a^2 + b^2 = c^2 + d^2. Let R* = all positive real numbers (including 0). Prove that there is a bijection between R* and the set of all equivalence classes for this equivalence relationship. Homework Equations...
  9. C

    How can i find a bijection from N( natural numbers) to Q[X]

    How can i find a bijection from N( natural numbers) to Q[X] ( polynomials with coefficient in rational numbers ). I can't find a solution for this. Can you please point me in the right direction ?
  10. S

    Complex Analysis - Proving a bijection on a closed disk

    Homework Statement For each w \in \mathbb{C} define the function \phi_w on the open set \mathbb{C}\backslash \{\bar{w}^{-1}\} by \phi_w (z) = \frac{w - z}{1 - \bar{w}z}, for z \in \mathbb{C}\backslash \{\bar{w}^{-1}\} \back. Prove that \phi_w : \bar{D} \mapsto \bar{D} is a...
  11. honestrosewater

    Prove this representation of naturals is a bijection

    Homework Statement Given s : \mathbb{N} \to \mathcal{P}(\mathbb{N}) s(n) = \{i | \exists (b_0,\ldots,b_k) \in \{0,1\}^{k+1} [n = \sum_j b_j2^j \,\wedge\, b_i = 1]\} show that s is a bijection between N and the finite subsets of N. In other words, if you express n as the sum of powers...
  12. S

    Proving the existence of a bijection.

    Hello all, I've recently used a property that seems perfectly valid, yet upon further scrutiny I could not come up with a way to prove it. Here is what I would like some help on. Given two sets X and Y and functions f and g mapping X into Y, with the property that f is injective and g is...
  13. L

    Can a bijection be extended to three dimensions?

    Can the definition of a bijection be extended to three dimensions? So for example, f(a, b) = c, where f : N X N \rightarrow N
  14. P

    Exhibit a bijection between N and the set of all odd integers greater than 13

    Homework Statement Exhibit a bijection between N and the set of all odd integers greater than 13 Homework Equations The Attempt at a Solution I didn't have a template for the problem solving. Please check if I did it in the right way? (The way and order a professor will like to see.)
  15. P

    (Real Analysis) Show the function is Bijection

    Homework Statement The problem and my attempt are attached. I am unable to solve the function for x. Homework Equations The Attempt at a Solution
  16. V

    Are Natural Bijections Only About Terminology in Category Theory?

    Seems like a silly question, but a search of the forum and Google and my online textbook yielded no results (*shakes fist at textbook writer). Please help?
  17. G

    Proving bijection between a region

    Homework Statement Prove bijection between the regions 0<x<1, 0<y<1, 0<u, 0<v, u+v<pi/2 Homework Equations x=sinu/cosv y = sinv/cosu The Attempt at a Solution We need to show that an inverse function exists to prove the bijection so obviously, (u,v) maps to one and only one...
  18. C

    Find explicit bijection from N to N x N

    Homework Statement Find a linear ordering of \mathbb{N} \times \mathbb{N} and use it to construct an explicit bijection f : \mathbb{N} \to \mathbb{N} \times \mathbb{N}. Homework Equations The Attempt at a Solution I know how to find a bijection by graphically by drawing the \mathbb{N} \times...
  19. B

    Continuous bijection that is not an embedding

    Hi: Just curious: a continuous function f:X-->Y ; X,Y topological spaces, can fail to be an embedding because it is not 1-1, or, if f is 1-1 , f can fail to be an embedding because, for U open in X f(U) is not open in f(X). Can anyone think of a "reasonable" example of the...
  20. J

    Show Explicit Bijection Between Sets (0,1)

    How can I show an explicit bijection between sets? anyidea? how about with (0,1)?
  21. X

    Help Urgently on Bijection, Injection, Subjection Functions

    Hi, I would like to say this is a great forum I found. This is my very first post yay =) I need help on these certain questions. 5. (10%) Given A = {1, 2, 3} and B = {a, b, c} (a) list in two-line notation all one-to-one functions from A to B; (b) list in two-line notation all onto functions...
  22. C

    Bijection between Orbit and Stabilizer

    So I know this is the orbit-stabilizer theorem. I saw it in Hungerford's Algebra (but without that name). So we want to form a bijection between the right cosets of the stabilizers and the orbit. Could I define the bijection as this: f: gG/Gx--->gx Where H=G/Gx f(hx)=gx h in H ^ Is that...
  23. K

    Injection, surjection, and bijection

    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...
  24. P

    Is There a Bijection Between Metric Spaces A and B?

    Hi, If A and B are two metric spaces and there exists two onto functions F and G such that F:A->B and G:B->A, is there a way to prove that there exists a bijection mapping A to B?
  25. R

    Bijection Proof for Sets and Functions - A Comprehensive Guide

    Homework Statement Let A and B be sets and let f: A \rightarrow B be a function. Define a function h: \mathcal{P} (B) \rightarrow \mathcal{P} (A) by declaring that for Y \in\mathcal{P} (B), h(Y)= \{ x \in A: f(x) \notin Y \}. Show that if f is a bijection then h is a bijection. The...
  26. T

    Prove Inverse of Bijection function

    Homework Statement Suppose f is bijection. Prove that f⁻¹. is bijection. Homework Equations A bijection of a function occurs when f is one to one and onto. I think the proof would involve showing f⁻¹. is bijective, by showing f⁻¹ is onto, and one to one, since f is bijective it is...
  27. J

    Inverse of continuous bijection

    Under which conditions is an inverse of a continuous bijection continuous? I'm not seeking for "the" answer. There probably are many. But anyway, I'm interested to hear about conditions that can be used to guarantee the continuity of the inverse. So far I don't know anything else than the...
  28. Q

    Complex analysis, finding a bijection

    Homework Statement Let Omega = C\((-inf,-1]U[1,inf)), find a holomorphic bijection phi:omega-->delta, where delta is the open unit disk Homework Equations Reimann Mapping Theorem Special Mapping formulas: can map wedges onto wedges, with deletion of real line from zero to infinity in...
  29. D

    Bijection in a complex number ring

    Homework Statement Show that the complex conjugation function f:C----->C (whose rule is f(a+bi)=a-bi) is a bijection Homework Equations A function is a bijection if it is both injective and surjective a function is injective if when f(a)=f(b) then a=b a function is surjective if for...
  30. J

    Proving Discontinuity of a Bijection: f-1:Y -> X

    Homework Statement Prove that if f:X -> Y is a discontinuous bijection then f-1:Y -> X is also discontinuous. Homework Equations N/A The Attempt at a Solution The contrapositive of this statement is that if f-1:Y -> X is continuous then f:X -> Y is continuous. Since f is...
  31. S

    Proving y' = 2ax + b is a Bijection

    Homework Statement Prove that the derivate of y = \frac {1} {2} ax^{2} + bx is a bijection, when a, b, x \in \Re The Attempt at a Solution y' = 2ax + b is a linear mapping, where a, b, x \in \Re . The mapping is \Re \rightarrow \Re . The mapping is an injection as each element in...
  32. R

    Find a Bijection [tex]\left[ 0,1 \right] \rightarrow \Re [/tex]

    Homework Statement Find a Bijection \left( 0,1 \right) \rightarrow R Homework Equations Detention of bijection The Attempt at a Solution let r be a number in the interval \left[ 0,1 \right] r = a b1c1b2c2 ... bncn where a, b,c are digits between [0, 9] f(r) = { - b1b2b3 ... bn . c1 c2...
  33. U

    Is There a Bijective Correspondence Between AxB and BxA?

    Homework Statement Show that there is a bijective correspondence of AxB with BxA. Homework Equations The Attempt at a Solution I am lacking the general understanding of this. Can I create a function g such that, g: (a in A, b in B) --> (b in B, a in A). If A and B are...
  34. S

    About the Jacobian determinant and the bijection

    Hello! I am having problems with the inverse function theorem. In some books it says to be locally inversible: first C1, 2nd Jacobian determinant different from 0 And I saw some books say to be locally inversible, it suffices to change the 2NDto "F'(a) is bijective".. How could these two be...
  35. N

    Question about injection, surjection, bijection, and mapping

    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...
  36. E

    Proving Bijection Between X and Y: Tips & Examples

    Prove that R ⊂ X x Y is a bijection between the sets X and Y, when R−1R= I: X→X and RR-1=I: Y→Y Set theory is a quite a new lesson for me. So I am not good at proving different connections, but please give me a little help with what to start and so.. I have read the book and I know what...
  37. S

    Proving the Equinumerosity of Infinite and Countable Sets

    Can you prove that \mathbf{R} and \mathbf{R}-\mathbf{Q} have same cardinality? One way would be to say that \mathbf{R}-\mathbf{Q} is not countable and must have cardinality <= \mathbf{R} and invoke the Continuum Hypothesis to conclude that its cardinality is aleph-1 same as that of...
  38. G

    Bijection between reals and irrationals

    Homework Statement Prove that |R \ |Q ~ |R ... the irrationals are equinumerous to the reals Homework Equations The Attempt at a Solution I can prove it using the Cantor Schroeder Bernstein theorem, but i was wondering if there is a clever way of constructing the bijection...
  39. J

    Is There a True Bijection Between the Natural Numbers and the Integers?

    There is a bijection between the natural numbers (including 0) and the integers (positive, negative, 0). The bijection from N -> Z is n -> k if n = 2k OR n -> -k if n = 2k + 1. For example, if n = 4, then k = 2 because 2(2) = 4. If n = 3, then k = -1 because 2(1) + 1 = 3. My problem...
  40. quasar987

    Bijection between Banach spaces.

    [SOLVED] Bijection between Banach spaces. Homework Statement Let E and F be two Banach space, f:E-->F be a continuous linear bijection and g:E-->F be linear and such that g\circ f^{-1} is continuous and ||g\circ f^{-1}||<1. Show that (f+g) is invertible and (f+g)^{-1} is continuous. [Hint...
  41. A

    Bijection between infinite bases of vector spaces

    I am reading "The linear algebra a beginning graduate student ought to know" by Golan, and I encountered a puzzling statement: Let V be a vector space (not necessarily finitely generated) over a field F. Prove that there exists a bijective function between any two bases of V. Hint: Use...
  42. M

    Topology: Bijection with Open intervals

    Good Morning, I am trying to prove that any 2 open intervals (a,b) and (c,d) are equivalent. Show that f(x) = ((d-c)/(b-a))*(x-a)+c is one-to-one and onto (c,d). a,b,c,d belong to the set of Real numbers with a<b and c<d. Let f: (a,b)->(c,d) be a linear function which i graphed to help me...
  43. phoenixthoth

    A Bijection from N to Q: Explicit Formula for Rational Numbers

    I am bored and feel like doing something useless today so I'm going to try to give an explicit formula that maps N to Q that is a one-to-one correspondence. If you want to waste some time, too, then feel free to post your functions or ideas towards that goal. Something that might be very...
  44. P

    Proving a Bijection between (0,1) and P(N)

    My proofs professor gave us this problem as a challenge, but I'm stuck, which is why I'm here: Let A,B sets Define set A to be (0,1) Define set B to be P(N), where N is the set of natural numbers, and P(N) is its power set. Question: Construct a bijection between (0,1) and P(N) or a...
  45. A

    Bijection, Injection, and Surjection

    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?
  46. G

    Can You Create a Bijective Function from the Reals to the Interval [0,1]?

    Hi, I'm trying to map all the reals into the interval [0,1]. I figured out that you can map all the numbers in the open interval (0,1) to all the reals by the function tan(pi(x-.5)) (so if I wanted a function from all the reals to (0,1) I could just take the inverse). But this problem is...
  47. G

    Ring, field, injection, surjection, bijection,

    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...
  48. J

    Prove f is a Bijection - Need Help

    Need help... If f(f(x))=x then prove f is a bijection.
  49. phoenixthoth

    Explicit Bijection from N to NxN

    for this post, let N = {1, 2, 3, ...}. Q: explicitly write down a closed form bijection with domain N and range N x N. (no need to prove its a bijection... just write down the formula.) this may lead you off track, but the way i did it was to first find a formula from N x N to N and then...
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