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.
Exponential polynomials are linear combinations of terms like ##t^ne^{at}\cos{bt}## and ##t^ne^{at}\sin{bt}##, where ##n## is a nonnegative integer, ##a## is real and ##b>0##. Like the proper rational functions, these are presumably subspaces of ##\mathbb R^{\mathbb R}##.
The proof goes like...
I have a hypothesis of which I wonder if it's sound. Perhaps you guys can advise me:
Suppose ##x_n\Rightarrow a_n## (logical implication) for some set X and set A. I think we have to assume a bijection.
Then, if ##a_m = False##, ##x_m## should be ##False##, right?
So, in case of a bijection...
Let f : N −→ R and f(x) = √ 9x
The domain is all natural numbers: {0, 1, 2, 3, ...}
The codomain is all real numbers.
The range i believe is [0, +infinity)
I believe that although the above is a function since every input of x provides a output that fits in our codomain. I also believe that...
So, I need to prove that ## (0,1] \thicksim (0,1) ##. Which means that I need to come up with some bijection from ##(0,1]## to ##(0,1)##. Now here is the outline of my function. I am going to use identity function for all irrationals. So, any irrational number in ##(0,1]## will be mapped to the...
How would one tackle this using the definition? (i.e. for some function ff that f(x)=f(y)⟹x=yf(x)=f(y)⟹x=y implies an injection and y=f(x)y=f(x) for all yy in the codomain of ff for a surjection, provided such x∈Dx∈D exist.)
One can solve the system of equations for x1x1 and x2x2 and that...
Hey! :o
Let $X$ be an infinite set and let $x\in X$. Show that there exists a bijection $f:X\to X\setminus \{x\}$. Use, if needed, the axiom of choice. To show that $f$ is bijective we have to show that it is surjective and injective.
The axiom of choice is equivalent to saying that, the...
Is the following (or, the following after any minor errors are corrected) a bijection from the unit square S=[0,1]X[0,1] to the line L=[0,1], or only an injection? If only an injection, are the excluded points in L countable?
[1] Let L be identified with the set of real numbers r, 0 ≤ r ≤ 1...
<Moderator's note: Moved from a technical forum.>
Hi PF,
I am learning how to prove things (I have minimal background in math). Would the following proof be considered valid and rigorous? If not any pointers or tips would be much appreciated!
Problem:
Prove that the notion of number of...
Homework Statement
Given two sets of Cartesian product
S=A1×A2...×An
P=(A1×A2...×An-1)×An
show that there exists bijection between the two sets.
Homework Equations
∀a1,a2:a1∈A1, a2∈A2: A1×A2=(a1,a2)
The Attempt at a Solution
let f be a function that maps
f: P → A1×A2...×An-1 where...
Homework Statement
Let p: E-->B be a covering map, let p(e_0)=b_0 and let E be path connected. Show that there is a bijection between the collection of right cosets of p*F(E,e_0) in F(B,b_0) (where p* is the homomorphism of fundamental groups induced by p and F(E,e_0),F(B,b_0) are the...
Theorem: Let ##A_1, A_2, ..., A_k## be finite, disjunct sets. Then ##|A_1 \cup A_2 \cup \dots \cup A_k| = |A_1| + |A_2| + \dots + |A_k|##
I will give the proof my book provides, I don't understand several parts of it.
Proof:
We have bijections ##f_i: [n_i] \rightarrow A_i## for ##i \in [k]##...
Homework Statement
Proof that: f has an inverse ##\iff## f is a bijection
Homework Equations /definitions[/B]
A) ##f: X \rightarrow Y##
If there is a function ##g: Y \rightarrow X## for which ##f \circ g = f(g(x)) = i_Y## and ##g \circ f = g(f(x)) = i_X##, then ##g## is the inverse...
I have been studying discrete mathematics for fun and I am kind of stuck on this bijection problem.
1. Homework Statement
I wanted to apologize in advance if i put this homework question in the wrong part of the forums. Discrete Math and much logic math is a computer science type math of...
Homework Statement
If G is a group and a ∈ G, let π: G--> G be the function defined as π(g) = ag, for all g ∈G.
a) Show that π is a bijection
b) Show that if π is an isomorphism, then a is the identity element of G.
Homework Equations
I think to show that pi is a bijection we have to show that...
Homework Statement
Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n.
Homework Equations
S_n = Symmetric set
≅ = isomorphism
Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...
I think my question is more appropriate here than in the computation section. My question is:
(In the context of inverse fast-fourier transforms and fast-fourier transforms)
Knowing ifft(fft(x)=x might be trivial as it is almost a definition; associating it with a domain ##t## is perfectly...
It is easy to construct a bijection between the open interval (0,1) and ℝ, and (if one isn't an intuitionist) it is easy to prove that there exists a bijection between [0,1] and ℝ, but is it possible to construct such a bijection between [0,1] and ℝ? Obviously it won't be continuous, but...
Homework Statement
I need to find a bijection between (0,1) and [0,1) in R. It can go in either direction since it is a bijection.
Homework Equations
I can't think of any equations at all!
The Attempt at a Solution
Something like f(x) = 1/[(1/x)+1] for x in A...
1. State whether or not f(x) = 2x mod n is a bijection for all odd n and prove it
Homework EquationsThe Attempt at a Solution
So I'm pretty sure it is a bijection over testing it for some odd n, and the fact that gcd(2,n) = 1, thus there must exist an inverse for 2x mod n. However, I realize...
Hello,
To measure the atomic force with an AFM. One can use the frequency shift of a cantilever. This change of frequency is linked to the atomic force by what we called the Franz J. Giessibl formula in the community.
z is the AFM tip-sample distance. frequency(z) is the change of...
I'd like to know if it is possible to define a bijection between the sets [0,1]^{\mathbb{Z}} and [0,1]^{\mathbb{N}}; \mathbb{N}^{\mathbb{N}} and \mathbb{Z}^{\mathbb{Z}}.
I tried to define a bijection between [0,1]^\mathbb{N} and [0,1]^\mathbb{Z} as follows: Take the bijection...
Homework Statement
Consider a bijection f = (A,B,F)
Show that f^(-1) (inverse of f) is a bijection from B to A and that for any element x of A we have
f^(-1)(f(x))=x
The Attempt at a Solution
For this proof can I use contradiction and the say f^(-1) is not a bijection from B to A or...
Homework Statement
Let K be any set and let F* be the set of all functions with domain K. Prove that card K < card F*.The Attempt at a Solution
I am first able to show that card K <= card F*, by creating an invertible function from K into F*.
let f: K -> F*
be defined so that if k is an...
I'm wondering whether my solution to this problem is correct, since the answer in the answer sheet says that this is provable, and I think I found a counterexample... Any help is appreciated :)
Homework Statement Prove or disprove: ##f:D->K##, where ##D,K != empty##, is a bijection iff there...
Let f:N-> Q be a bijection. I want to show that this is uniformly continuous on N. (N is the set of natural numbers, Q the rationals). My first thought was to use induction. Since every point in N is an isolated point, then f is continuous on N.
Let N1=[1,a_1], where a_1 is a natural number...
Can anyone explain to me how to do these types of questions? I have the answers but I don't understand it.
The function f: N -> N, f(n) = n+1 is
(a) Surjection but not an injection
(B) Injection but not a surjection
(c) A Bijection
(d) Neither surjection not injection
The answer is B...
1. Homework Statement .
Let A and B be disjoint sets, A infinite and B enumerable. Prove that there exists a bijection between AuB and A.
3. The Attempt at a Solution .
I have an idea of how to prove this statement, but I got stuck in the middle, so here is what I've done:
There are just...
Homework Statement
Prove that |AB\cupC|=|ABx AC| by demonstrating a bijection between the two sets.
Homework Equations
Two sets have equivalent cardinality if there is a bijection between them/
The Attempt at a Solution
Essentially I can prove that there is a function from...
Homework Statement
I want to prove this proposition:
Let f: M \rightarrow N be a uniformly continuous bijection between metric spaces. If M is complete, then N is complete.The Attempt at a Solution
I have a 'partial' solution, whose legitimacy hinges upon a claim that I am unable to prove...
Homework Statement
Let X be a set. Suppose that f is a bijection from p(X) to p(X) such that f(A)\subseteq f(B) iff A\subseteq B for all subsets A,B of X.
Show that there is a bijection g from X to X such that for all A\subseteq X one has f(A)=g(A).
Homework Equations
p(X) is the power...
Homework Statement
let V be finite -dimentional and T:V->V*(V* is the dual space of V with same dimension as V) ,let ei be the bases of V,e^i be the bases of V*,consider the linear bijection :K:V->V* defined K(ei)=e^i,show that this bijection depends on the original choice of basis.
2. The...
For two sets X and Y let X^Y be the set of functions from Y to X.
Prove that there is a bijection between (X x Y)^Z and X^Z x Y^Z.
Attempt: I could not get any further from that "there must be a function S with S(f)=g and S(f')=g for any g, f' in X^Z x Y^Z, and where f is in (X x Y)^Z."
1. The problem statement, all variables and given/known data
Let (G,*) be a group, and denote the inverse of an element x
by x'. Show that f: G to G defi ned by f(x) = x' is a bijection,
by explicitly writing down an inverse. Given x, y in G, what is
f(x *y)?
Homework Equations...
Homework Statement
I need to prove that [0,1) and (0,1] have the same cardinality. My question is, do I have to define a function from [0,1) → (0,1] in order to show a correspondence or is there another method?
Thanks.
Homework Statement
Let X= \{a,b,c\} and Y= \{d,e\}. Write down and explicit bijection N_{|X×Y|} → X×Y
The Attempt at a Solution
Well I came up with the easiest method, just giving one value to each member of N_{|X×Y|} so I was just wondering whether there is another way of doing it not by...
Homework Statement
If f:A->B is a bijection and g:B->C is a bijection, show that g\circf is a bijection from A->C.Homework Equations
A function is a bijection from A to B when it is both a surjection and an injection from A to B.The Attempt at a Solution
Suppose f is a bijection from A to B and...
Homework Statement
I was given a pdf document containing questions that require me to prove set rules. However, the third question (the one that starts at the bottom of the first page and runs into the second page) is giving me problems. I might be able to prove it if he wants a proof by...
Homework Statement
With X and Y being sets, I need to show that there is a bijection between the set of functions (X->P(Y)) and the set of relations P(X x Y) where P(..) is the power set symbol.
Homework Equations
P(Y) = {Z | Z subset of Y}
The Attempt at a Solution
I know a bijection...
My analysis text defines inverse functions only for bijections.
But y = e^{x} is not bijective, so according to my book it's inverse ( ln x ) wouldn't be defined? Am I missing something or is my textbook just plain wrong?
I use the text by Bartle and Sherbert.
Thanks!
BiP
Homework Statement
prove there is no continuous bijection from the unit circle (the boundary; x^2+y^2=1) to R
Homework Equations
The Attempt at a Solution
is this possible to show by cardinality? since if two sets have different cardinality, then there is no bijection between...
a function f, that maps from the Cartesian Product of the positive integers to the positive integers. where
f(x,y) = 2^(x - 1) * (2y - 1).
I have to show that this function is both one-to-one and onto. I started trying to prove that it is onto, showing that there exists an n such that...
Homework Statement
Question 1:
Prove that the cardinality of R (the set of all real numbers)is the same as the cardinality of R-{0} by constructing a bijective function from R to R-{0}
Question 2: Let A be the infinite sequence of binary numbers as follows:
A={(a1,a2,a3...)|ai= 0or 1 for...
Homework Statement
Let S1 = {a} be a set consisting of just one element and let
S2 = {b, c} be a set consisting of two elements.
Show that S1 × Z is bijective to S2 × Z.
Homework Equations
The Attempt at a Solution
So I usually prove bijectivity by showing that two sets are...
Homework Statement
here's the problem:
Let A and B be n x n matrix with coefficient in K (any field), let Mn(K) be the set of all n x n matrix with coefficient in K . T is a linear map defined like this
T : Mn(K)---> Mn(K)
T(Y) = AYB
what are the necessary conditions for T to be a...
Let A,B, and C be non-empty sets. A and B are bijective.
Prove that the power set of A is bijective to the power set of B.
I understand how to prove bijection but can't figure out how to apply this to power sets and can't find any info on this subject.
Let A1, A2, T be non-empty sets such that A1 is bijective to A2.
Show that A1 × T is bijective to A2 × T
So far I've been able to show that for any b where b is an element of A2, there must be some a within A1 such that f(a) = b. I've been able to do the same for proving injectivity...
How can I prove this:
Let A, B, and C be non empty sets. If A is bijective to B, then A x C is bijective to B x C.
also if A and B are bijective Power set of A is bijective to Power set of B
and finally Fun(A,C) is bijective to Fun(B,C)