What is Cardinality: Definition and 173 Discussions
In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set
A
=
{
2
,
4
,
6
}
{\displaystyle A=\{2,4,6\}}
contains 3 elements, and therefore
A
{\displaystyle A}
has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers.
The cardinality of a set is also called its size, when no confusion with other notions of size is possible.
The cardinality of a set
A
{\displaystyle A}
is usually denoted

A

{\displaystyle A}
, with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set
QUESTION:
If A is a finite set, its cardinality, o(A), is the number of elements in A. Compute
(a) o(A) when A is the set consisting of all fivedigit integers, each digit of which is 1, 2, or 3.
(b) o(B), where B = {x ∈ A : each of 1,2 and 3 is among the digits of x} and A...
Problem: Find the cardinality of the set ## A = \{f \in \Bbb N \to \Bbb N. \forall n\leq m .f(n) \geq f (m) \} ##.
I know that ## A \subseteq P(\Bbb N \times \Bbb N) ## implies ## A \leq P(\Bbb N \times \Bbb N) =  P(\Bbb N)  = \aleph ##. So I have a feeling that ## \aleph \leq A ##...
Hi everyone,
I have a quick question about Aleph numbers. Are they even possible? By containing infinity to a finite set, isn’t that essentially disproving the infinity in the first place? Can they be used in an actual scenario’s, or are they just purely hypothetical? Can they be used to...
Suppose we construct a set, adding at each step a polynomial number of elements.
My impression that after we do countably infinite number of steps, the set will have countably infinite cardinality.
But what happens if we add exponential number of elements each step?
For instance, on step 0 we...
By ZFC, the minimal set satisfying the requirements of the axiom of infinity, is the intersection of all inductive sets.
In case that the axiom of infinity is expressed as
∃I (Ø ∈ I ∧ ∀x (x ∈ I ⇒ x ⋃ {x} ∈ I))
the intersection of all inductive sets (let's call it K) is defined as
set K = {x...
Mentor note: In this thread I (Mark44) have edited "cardinal" to "cardinality." In English, we talk about the "cardinality of a set," not the "cardinal of the set."
Given A a set of n elements  note A its cardinal and P(A) its powerset. Let A1, A2... Ak, be k subsets (not empty) of A.
What...
Physics speaks of a set S of N "indistinguishable particles", giving the set S a cardinality but forbidding any equivalence relation that can distinguish between two particles. Is this terminology inconsistent with the mathematical definition of cardinality?
Suppose ##S## is a set with...
Given a Structure , is the Set of True Sentences (which I think is called the full theory generated by the Structure) with, say, countable symbols. Is there always a bijection between the true sentences (Semantic ) vs the Theorems (Syntactic)? I believe this depends on the existence of a model...
The interval ##[0,1]## of real numbers has a nonzero measure. The set of all rational numbers in the interval ##[0,1]## has zero measure. But there are also sets that are somewhere in between, in the sense that their measure is neither zero nor nonzero. They are sets for which measure is not...
The PF spellchecker suggests replacing the mathematical term "cardinality" with "carnality". Are these terms synonymous?
Hmm ... finite carnality ... infinite carnality ... different levels of infinite carnality ...
First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model)
Given that, I would like to...
I am working on a proof problem and I would love to know if my proof goes through:
If $A, B$ are sets and if $A \subseteq B$, prove that $A \le B$.
Proof:
(a) By definition of subset or equal, if $x \in A$ then $x \in B$. However the converse statement if $x \in B$ then $x \in A$ is not...
Homework Statement
In a finitedimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list.
It's quite long :nb), hope you guys read through it. Thanks! :smile:
Homework Equations
N/A
The Attempt at a Solution...
Can (0,1)\subset\mathbb{R} be divided into an infinite set S of nonempty disjoint subsets? It seams like any pair of points in different subsets of the partitioning must have a finite difference, and so there must be some smallest finite difference overall, d where S \leq 1/d. Can someone...
I'm having trouble following one part of a proof.
Proposition: For any covering map ##p: X>Y##, the cardinality of the fibers ##p^{1}(q)## is the same for all fibers
Proof: If U is any evenly coverd open set in ##X##, each component of ##p^{1}(q)## contains exactly one point of each fiber...
In the MWI, are the number of universes in the multiverse countable or uncountable? It seems like if all possibilities happen, then that is like the power set, which has uncountable cardinality. Or maybe a Cantor diagonalization argument can be used on the discrete sequence of events over the...
Homework Statement
Prove that the cardinality of ##P(S)## is greater than the cardinality of S, where S is any set.
Homework EquationsThe Attempt at a Solution
It would seem that we could simply define ##T: S \rightarrow P(S)## such that ##T(s) = \{s \}##. This is clearly an injection, so...
Homework Statement
##\mathbb{R} \setminus C \sim \mathbb{R} \sim \mathbb{R} \cup C##.
Homework EquationsThe Attempt at a Solution
I have to show that all of these have the same cardinality. For ##\mathbb{R} \cup C \sim \mathbb{R}##, if ##C = \{c_1, c_2, ... c_n \}## is finite we can define ##...
To show that two sets have the same cardinality you have to show that there is a bijection between the two. Apparently, one bijection from [0,1] to (0,1) is ##
f(x) = \left\{
\begin{array}{lr}
1/2 & : x = 0\\
\frac{1}{n+2} & : x = \frac{1}{n}\\
x & : \text{any other...
According to this page: https://en.wikipedia.org/wiki/Cantor's_theorem
It says: "Cantor's theorem is a fundamental result that states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself."
Furthermore, it says: "Cantor's...
Hi,
I hoping someone might be kind enough to possibly tell me where I have made an error :)
I'm more of a recreational maths person, lol  and I'm trying to make a scheme that 'maps' any decimal number to a natural one.
The method I have come up with is a bit odd, I'm hoping it works but still...
To select an element from countably infinite set (list set of integers) you need to provide finite amount of information. To specify an element in continuum in general case you have to provide infinite amount of information: any real number is specified as countableinfinite number of digits. So...
Hey! :o
We have the set $A=\{a_1, a_2, \ldots \}$, the $a_i$'s might be finitely or infinitely many.
We have that $\mathbb{Q}(A)=\left \{\frac{f(a_1, \ldots , a_n)}{g(a_1, \ldots , a_n)} : f,g\in \mathbb{Q}[x_1, \ldots , x_n], g\neq 0, a_1, \ldots , a_n\in A, n\in \mathbb{N}\right \}$. We...
Hello all.
I have a question concerning following proof, Lemma 1.
http://planetmath.org/allbasesforavectorspacehavethesamecardinalitySo, we suppose that A and B are finite and then we construct a new basis ##B_1## for V by removing an element. So they choose ##a_1 \in A## and add it to...
Is there a consensus on the cardinality of the infinite number of universes in the Multiverse?
Is it countable or more than countable?
Is it the same in different theories?
Cardinality of the set of binaryexpressed real numbers
This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis.
1. Counting...
Hello,
At my exam I had to proof the title of this topic. I now know that it can easily be done by making a bijection between the two, but I still want to know why I didn't receive any points for my answer, or better stated, if there is still a way to proof the statement from my work.
My work...
Sets with cardinality ##2^{\aleph_0}##, that is, with cardinality of the set of real numbers, obviously have many applications in other branches of mathematics outside of pure set theory. For example, real any complex analysis is completely based on such sets.
How about higher cardinality? Is...
Hi! (Wave)
Find the cardinal number of $C(\mathbb{R}, \mathbb{R})$ of the continuous real functions of a real variable and show that $C(\mathbb{R}, \mathbb{R})$ is not equinumerous with the set $\mathbb{R}^{\mathbb{R}}$ of all the real functions of a real variable. That's what I have tried: We...
Hello! (Wave)
Suppose that $X$ contains a countable set. Let $b \notin X$. Show that $X \sim X \cup \{b\}$.
Prove that in general if $B$ is at most countable with $B \cap X=\varnothing$ then $X \sim X \cup B$.
Proof:We will show that $X \sim X \cup \{b\}$.
There is a $\{ a_n: n \in \omega \}...
O.k. I am seriously confused... Not being to good at math but nevertheless interested in set theory, infinity, etc. I started reading Mary Tiles, The Philosophy of Set Theory (Dover edition). I particularly wanted to know more about the relation between infinite ordinals and cardinality, but...
Hello! (Smile)
If we want to show that the sets:
$$A=\{ 3X^2 X \in \mathbb{Z}_p \}\ \ \text{ and } \ \ B=\{ 75Y^2 Y \in \mathbb{Z}_p\}$$
have the same cardinality, could we take the bijective function $f$ such that $f(x)=\frac{75x}{3}$ ? Or am I wrong? (Thinking)
Consider N random variables X_{n} each following a Bernoulli distribution B(r_{n}) with 1 \geq r_{1} \geq r_{2} \geq ... \geq r_{N} \geq 0. If we make following assumptions of sets A and B:
(1) A \subset I and B \subset I with I=\{1,2,3,...,N\}
(2) A \cap I_{1} \geq B \cap I_{1} with...
Dear all,
I have a question attached related to both probability and cardinality. Let me know if my formulation of the problem is nonrigorous or confusing. Any proof or suggestions are appreciated.Thank you all.
The question follows.Consider a set \(I\) consists of \(N\) incidents...
Hello everyone
I was given a question in a homework (this is not a homework thread though as I have submitted it)
is was to :
Show that there is no infinite set A such that A < Z+ =ℵ0.
I thought of it and tried to work my way out and came up with those proofs , which I am not quiet sure...
Let $I$ be an interval and $A_{n}$ be the set of $k/n$ where $k$ is an integer.
Prove that $I$ is the limit as $n$ tends to infinity of $\frac{1}{n}(IA_{n})$ where $IA_{n}$ denotes intersection.
My plan was to split it up into cases for the different type of intervals and come up with...
Define
F(A) = The set of all finite subsets of A
Seq(A) = The set of all finite sequences with elements from A
Let A be an infinite set (not necessarily countable).
I want to prove the following lines.
1. Card seq(A) \le Card(A^\omega)
2. Card A = Card seq(A) = Card F(A)
I know that we can easily construct a set whose cardinality is strictly greater than that of the set of real numbers by taking P(\Re) where P denotes the powerset operator. But as far as I am aware there aren't really any uses for this class of sets (up to bijection), or any intuitive ways of...
Homework Statement
Let S be the set of functions from a set A to {0,1} Prove that P(A)= S
Homework Equations
P(A) is the power set of A
The Attempt at a Solution
I have no idea how to do this... If A is finite then A has n elements, and we can write out the elements from one to...
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...
Homework Statement
Prove that the open interval (1,3) and the closed interval [1,4] have the same cardinality.
Homework Equations
The Attempt at a Solution
I have to prove bijection.
The injective part is obvious.
Say, A =(1,3) and B =[1,4]
f: A → B
f(x) = x
It's...
I have faced the following problem recently:
We have a sequence A of M consecutive integers, beginning at A[1] = 1: 1,2,...M (example: M = 8 , A = 1,2,3,4,5,6,7,8 )
We have the set T consisting of all possible subsequences made from L_T consecutive terms of A, which do not overlap. (example...
Hey! :o
I am looking at an exercise and I got stuck...
$n\epsilon \mathbb{N},n>1$
$φ(n)=\{1 \leq k \leq n :$ the greatest common divisor of $k$ and $n$ is $1\}$
I am asked to find $φ(n)$,but I don't know how...
Am a bit confused about the meaning of cardinality. If ## A \subseteq B ##, then is it necessarily the case that ## A \leq B ##?
I am thinking that since ## A \subseteq B ##, an injection from A to B exists, hence its cardinality cannot be greater than that of B?
But this cannot be...
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...
Hey everyone, I have three problems that I'm working on that are review questions for my Math Final.
Homework Statement
First Question: Determine if R is an equivalence relation: R = {(x,y) \in Z x Z  x  y =5}
and find the equivalence classes.
Is Z  R a partition?
Homework...
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?