Cardinality Definition and 174 Threads

Prove that the interval A = [0 , 2) has the same cardinality as the set B = [5 , 6) U [7 , 8) by constructing a bijection between the two sets Attempt: x ↦ x + 5 for x ∈ [0 ; 1) x ↦ x + 6 for x ∈ [1 ; 2) What to do next?

Homework Statement . Let ##f:ℝ→ℝ## such that f is piecewise linear, which means, for every ##x \in ℝ##, there is an ##ε>0## such that f restricted to ##[x-ε,x]## and restricted to ##[x,x+ε]## are linear functions. Find the cardinality of ##A##={##f:ℝ→ℝ## / ##f## is piecewise linear} The...

Homework Statement Let G be a finite group where H and K are subgroups of G . Prove that |HK|=\frac{|H||K|}{|H \cap K|} . Homework Equations set HK=\{x\in G| x=st, s\in H and t\in K\}The Attempt at a Solution I am a bit lost with this problem. What I did was break this proof into...

Is 0 I am told. Is this an axiom, or can it be proven?

I saw the below statement which is intuitively correct: If a set has cardinality m then none of its subsets has cardinality greater than m. Is it necessarily true for a infinite set case?

I have seen a lot of examples of sets with same cardinality as the natural numbers. For instance the even numbers or the cartesian product. In any case the proof amounted to finding a way of labeling the elements uniquely. But I am curious - can anyone give me an example of a set, where this...

Homework Statement What is the cardinality of the set of all numbers in the interval [0, 1] which have decimal expansions with a finite number of non-zero digits?Homework Equations The Attempt at a Solution I say its still c? Am I correct, there is no way I can pair this set with the natural...

i have solved these problem just want to make sure I'm on the right track. 1. Say the football team F, the basketball team B, and the track team T, decide to form a varsity club V. how many members will V have if $n\left(F\right)\,=\,25,\,n\left(B\right)\,=\,12,\,n\left(T\right)\,=\,30$ and no...

Homework Statement What is the cardinality of the set of all functions from N to {1,2}? Homework Equations The Attempt at a Solution I know the cardinality of the set of all functions coincides with the respective power set (I think) so 2^n where n is the size of the set. The...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Homework Statement Let S be the set of all functions mapping the set {√2, √3, √5,√7} into Q. What is the cardinality of S?Homework EquationsThe Attempt at a Solution I have been stuck staring and trying to think of something to figure out this question. This is the idea i have: let U = { all...

What is the cardinality of R2? Seems like it should be a fairly simple to explain, yet I'm stuck beyond belief. Attempt: R2 = R x R Now we have shown that the |R| = | [0,1] | but then when I think of possibly combining that fact I'm still somewhat in the same place. How do I...

Here is the question: Here is a link to the question: Cardinality of Sets Homework Problem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Suppose that A and B are finite sets. What is |P(AxB)|? Meaning what is the cardinality of the power set of a cartesian product of the sets A and B. Homework Equations |AxB|=|A| * |B| since A and B are finite sets Power set of a set is the set of all subsets of...

I have to prove that the cardinality of the set of infinite sequences of real numbers is equal to the cardinality of the set of real numbers. So: A := |\mathbb{R}^\mathbb{N}|=|\mathbb{R}| =: B My plan was to define 2 injective maps, 1 from A to B, and 1 from B to A. B <= A is trivial, 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...

What would the cardinality of the set of all ordinal numbers be? Is it even known or does the question even make sense in the case of such a weird, almost paradoxical set?

Two (related) questions: (A) If I understand correctly (no guarantee to that), in an Everett-type Many-Worlds-Theory of Quantum Mechanics, every probability amplitude is associated with a world. This would mean, for a single particle, that there would be as many worlds ("be" in the sense of a...

I was wondering. Has anyone ever been able to build a beth-2 set? What does it look like? What could it possibly look like?

Homework Statement Prove that if A,B, and C are nonempty sets such that A \subseteq B \subseteq C and |A|=|C|, then |A|=|B| The Attempt at a Solution Assume B \subset C and A \subset B (else A=B or B=C), and there must be a bijection f:A\rightarrowC...

This is not a homework question ... If two vector spaces, say V and W, have equal cardinality |V|=|W| ... do they then have the same dimension? That is dim(V)=dim(W)? I am struggling with making this call one way or the other. This is no area of expertise for me by any means so I know I...

I proved that [0,1) has the same cardinality as (0,1], by defining a function and then checking injectivity/surjectivity. I proved [0,1] has the same cardinality as (0,1), by defining a function and showing it has an inverse. I now have to prove that (0,1] has the same cardinality as [0,1]...

There are no actual paradoxes in mathematics (we hope). There are only things that appear to be paradoxes to fallible intuitions. Here is one that bothers me. There can be no translation invariant probability measure on the non-negative integers. Yet it is possible to imagine a general class...

I'm going to construct an ordered set, and I'd like to ask some questions about it; and in particular consider coding problems about this set and sets in general (Turing tape encoding). Start with: A={ } And, allow a mutable temporary set, initialized with: T={ } and an iterator: set n=0 For...

I think the cardinality of the set M of all 1-1 mappings of the integers to themselves should be the same as the cardinality of the real numbers, which I'll denote by \aleph_1 . My naive reasoning is: The cardinality of all subsets of the integers is \aleph_1 . A subset of the integers...

The notation has me a bit confused... Heres my logic for the P({1}) on the inside {EmptySet, {{1}}} reason being, you always include the empty set, {1} is a part of the set. The cardinality is two You have the set: {EmptySet, {{1}}}, and now you have to consider the outer "P" the...

I was doing one of the proofs for my abstract algebra class, and we had to prove that the cardinality of the image of G, [θ(G)] is a divisor lGl. I'm trying to intuitively understand why G and it's image don't necessarily have the same cardinality. I'm thinking it's because there isn't...

Homework Statement Suppose \mathbb{Q},\mathbb{R} are the set of all rational numbers and the set of all real numbers, respectively. Then what is |\mathbb{R} \backslash \mathbb{Q}|?Homework Equations |\mathbb{Q}| = |\mathbb{Z^{+}}| < |P(\mathbb{Z^{+}})| = |\mathbb{R}|The Attempt at a Solution I...

Homework Statement Let NN be the set of all functions from N to N. Prove that |NN|=c Homework Equations The Attempt at a Solution I can prove that the set of all functions from N to {0,1} has cardinality of the continuum, but i can't generalise it. Any help would be appreciated.

Cardinality of the Preimage f^{-1}(y) of f:X-->Y continuous? Hi, All: Let X,Y be topological spaces and f:X-->Y non-constant continuous function. I'm curious as to whether it is possible for the fiber {f^{-1}(y)} of some y in Y to be uncountable, given that the fiber is discrete (this...

Homework Statement Show by induction that if the finite sets A and B have m and n elements, respectively, then (i) A X B has mn elements; (ii) A has 2m subsets; (iii) If further A \cap B = \varphi, then A \cup B has m+ n elements. NOTE : I am only interested in the (iii) section of...

Homework Statement If A and B are sets we say that |A|≤|B| if and only if there exists a one-to-one function f:A→B. Prove that if A and B are sets such that A\subseteqB , then |A|≤|B|. Homework Equations Our text does not define this, so the definition comes from my class...

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...

Hi - I've got the following question but can't find any concrete information in my books on how to answer it and I'm slightly confused: {x ε R : 2≤x≤3 } and {x ε R : 2≤x≤5 } Do they have the same cardinality? My understanding of this is if you can find a mapping that satisifies a bijection...

Homework Statement Show that the number of distinct solutions of a system of linear equations (in any number of equations, and unknowns) over the field Zp is either 0, or a power of p. The Attempt at a Solution First off, I was wondering whether there is any difference between...

Hello, Given a Brownian Motion process B(t) for 0≤t≤T, we can write it more explicitly as B(t,ω) where ω\inΩ, where Ω is the underlying sample space. My question is: what is the cardinality of Ω. I.e. what is |Ω|? My thoughts are that it is an uncountable set, based on the observation...

Homework Statement I need to show that [0,1] and [0,2] have the same cardinality by giving a formula for a function that is bijective. Aren't there a number of functions that can fit this description? Can I then use any one? I'm a little confused, my teacher didn't really elaborate much upon...

Hello, I was wondering this, what is the cardinality of the set of all finite subsets of the real interval [0,1] It somehow confuses me because the interval is nonnumerable (cardinality of the continuos \mathfrak{c}), while the subsets are less than numerable (finite). It is clear that it has...

From a pdf textbook: Example (infinite sets having the same cardinality). Let f : (0, 1) → (1,∞) be defined by f(x) = 1/x. Then f is a 1-1 correspondence. (Exercise: prove it.) Therefore, |(0, 1)| = |(1,∞)|. Exercise. Show that |(0,∞)| = |(1,∞)| = |(0, 1)|. Use this result and the fact that (0,∞)...

Homework Statement Determine whether or not the set is countable or not. Justify your answer. The set Bn of all functions f:{1,2,...,n}\rightarrowN, where N is the natural numbers. Homework Equations 1.)A countable union of countable sets is countable 2.)A finite product of...

Homework Statement F(\mathbb{Q},\mathbb{R}) is the set of maps from \mathbb{Q} to \mathbb{R}. Then show that F(\mathbb{Q},\mathbb{R}) and \mathbb{R} have same potency (cardinal number?).. Homework Equations The Attempt at a Solution I am no tsure but I think I need to...

Does anyone happen to know what the cardinality of the set of ordinal number (transfinite and otherwise) is? A simplified proof would also be much appreciated. Recently I have been very interested in transfinite numbers and the logically gorgeous proofs involved :D

Homework Statement Hi! I want to show that lXl<lYl implies lXl\inlYl where lXl and lYl are some cardinal numbers of two sets X and Y and the ordering < is defined on cardinal numbers . Homework Equations The Attempt at a Solution I tried to solve it by myself as follows: lXl...

S is the set containing all 2x2 invertible matrices such that the entries come from the the set {0,1,2}. What is the cardinality(number of elements) of this set? I got 50. Is this correct? What is the best way to go about solving this problem?

In the https://www.physicsforums.com/showthread.php?t=507003" , we have looked at various types of infinities. In the last section of that post, we have said when we regarded two sets to have equal size (or equal cardinality). We will now flesh out this concept a bit. Comparing sizes of sets...

How do you prove that if \textrm{card}(X)\leq\textrm{card}(Y) is not true, then \textrm{card}(X)\geq\textrm{card}(Y) must be true? In other words, if we know that no injection X\to Y exists, how do we prove that an injection Y\to X must exist? This is not the same thing as what...

How do you prove that there does not exist a set X such that \textrm{card}(X) < \textrm{card}(\mathbb{N}) but still n < \textrm{card}(X),\quad \forall\;n\in\mathbb{N} ----------------- edit: I proved this already. No need to answer... ------------------ I came up with a new question...

correct me if I'm wrong, but the set of Natural numbers and the set of all positive even numbers have the same number of elements, the same cardinality, right? So there would have to be a bijective function between the two, correct? If we go from f:N->N then the function is not surjective...

Homework Statement Let U and V both have the same cardinality as R (the real numbers). Show that U\cupV also has the same cardinality as R. Homework Equations The Attempt at a Solution Because U and V both have the same cardinality as R, I that that this means \exists f: R\rightarrowU that is...

Let V be a vector space over an infinite field $\mathbf{k}$. Let \beta be a basis of V. In this case we can write V\cong \mathbf{k}^{\oplus \beta}:=\bigl\{ f\colon\beta\to \mathbf{k}\bigm| f(\mathbf{b})=\mathbf{0}\text{ for all but finitely many }\mathbf{b}\in\beta\bigr\}...