3
votes
2answers
27 views

cardinality of infinite sets with cartesian product

claim: $A,B,C,D$ are infinite if $|A\times B|=|C\times D|$ then $|A|=|C|$, $|B|=|D|$ , prove or give a counter example. So imo, the claim is false, using $A=D=\mathbb{R}$ , $B=C=\mathbb{N}$ , is it ...
1
vote
1answer
44 views

show that for an infinite cardinal $k$, $k + k = k$

Show that for an infinite cardinal $k$, $k + k = k$ So far I have that $k + k = 2k$ Is it possible to somehow show that $2k = k$? I've been trying to understand some cardinal arithmetic, and I ...
3
votes
2answers
39 views

Question about partitions in intervals of the real numbers.

I have to prove the following: Let $ \mathcal{D} $ be a partition of $ \mathbb{R} $ in intervals of any kind, except intervals containing a single element. Prove $ \mathcal{D} $ is countable. ...
0
votes
2answers
48 views

Cardinality of two sets cross-multiplied

Let $A$ and $B$ be sets. Prove that $ \#(A \times B) = \#(B \times A)$. What I have done: There exist an element $m$ in $A$ such that the element also exists in $B$. If $\#A = \#B$, then $\#B = ...
3
votes
1answer
75 views

Cardinality of the set of permutations of a set $ A $

I've some trouble calculating the cardinality of the set of the permutations of a given set $ A $. For notational purpose let $ k = |A|$ and define $ P_A = \{ f : A \to A | f \text{ is a bijective ...
2
votes
1answer
79 views

prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality

I have to prove that $\Bbb{Z \times ((0,1]\cap Q)}$ and $\Bbb Q$ have the same cardinality. I think I have bijective function($f(\langle a,b\rangle)=a-b$) between thse sets, but I don't know how to ...
3
votes
1answer
218 views

Find the cardinality of these sets

Question from my homework im struggling with Find the cardinality of these sets: 1) the set of all sequences of natural numbers 2) the set of all arithmetic series (difference between 2 ...
2
votes
1answer
55 views

Elementary set theory, Cantor-Bernstein-Schröder usage, check my proof

I have a question, I was asked to show that $[0,1]$ and $\mathbb R$ are of equal cardinality using the Cantor-Bernstein-Schröder theorem. I would just like some feedback, if I solved it correctly: ...
3
votes
1answer
62 views

Homework - countable infinity

I'm trying to solve 2 problems, but I'm having some issues and would appreciate help. Here are the questions and what I thought could be done: 1) A is the set of all series of numbers, where in an ...
0
votes
2answers
74 views

Proove formally that |N| = | N union a finite set |.

I'd like to show that the cardinality of $\mathbb{N}$ is the same as the cardinality of $\mathbb{N}$ union some other finite set (disjoint from $\mathbb{N}$) For e.g show that : $|\mathbb{N}|= | ...
1
vote
2answers
108 views

Cardinality and bijections

i am following a course in axomatic set theory. We are talking about bijections, injections, Schröder-Bernstein-theorem, etc. at this moment. I want to make te following exercises: Prove: ...
2
votes
1answer
82 views

Proving that a particular subset of $\mathbb{R}$ is countable.

I've been trying to solve this problem for a few days and I feel like I'm missing something big. Let $ X \subseteq \mathbb{R}_{>0} $ so that there's a $C > 0$ such that for every finite ...
3
votes
1answer
139 views

Finite and infinite sets, cardinality question

Suppose there are infinite sets $A$, $B$ and $C$ such that $$|A| = |B| = |C| = |\mathbb{N}|\\ |D| = |\mathbb{R}|$$ and the finite set $E$ Give an example for the following (using the sets above). In ...
2
votes
1answer
94 views

Similar to Fodor lemma

Let $\lambda>\aleph_0$ be a regular cardinal such that $S \subseteq \lambda$ is not a stationary subset. Prove that there exists a regressive function $f:S \to \lambda$ such that ...
2
votes
1answer
57 views

Example of a $\kappa$-long sequence of disjoint club subsets of regular cardinal $\kappa$

I'm self-studying set theory and got stuck on this exercise: Let $\kappa$ be a regular cardinal. Give an example of a sequence $\langle C_\alpha\mid\alpha<\kappa\rangle$ such that $C_\alpha$ is ...
0
votes
2answers
166 views

Closed, unbounded subset of a cardinal.

I missed two lectures in my set theory course, and now I don't understand the homework problems. One is this: let $\kappa$ be a regular uncountable cardinal. Show that the following sets are closed ...
3
votes
1answer
64 views

Why are there $2^{\aleph_0}$ injections from $\omega$ to $\omega_1?$ [duplicate]

I have to prove that there are $2^{\aleph_0}$ injections from $\omega$ to $\omega_1.$ I can see that there is a bijection between this set and the set of pairs: (permutation of $\omega$, infinitely ...
0
votes
2answers
210 views

If $A$ and $B$ are denumerable sets, and $C$ is a finite set, then $A \cup B \cup C$ is denumerable

I have a statement here I wish to prove and I would love some help on it :) If $A$ and $B$ are denumerable sets, and $C$ is a finite set, then $A \cup B \cup C$ is denumerable Here is my ...
3
votes
3answers
610 views

Let $A$ be any uncountable set, and let $B$ be a countable subset of $A$. Prove that the cardinality of $A = A - B $

I am going over my professors answer to the following problem and to be honest I am quite confused :/ Help would be greatly appreciated! Let $A$ be any uncountable set, and let $B$ be a countable ...
3
votes
3answers
458 views

Cardinality of a set A is strictly less than the cardinality of the power set of A

I am trying to prove the following statement but have trouble comprehending/going forward with some parts! Here is the statement: If $A$ is any set, then $|A|$ $<$ $|P(A)|$ Here is what I ...
3
votes
3answers
99 views

$P(A)$ and $2^A$ are numerically equivalent

Whilst reading some notes on the cardinality of infinite sets, I got to this question which has been bugging me for a while. Help would be greatly appreciated! For every nonempty set A, the sets ...
0
votes
1answer
46 views

X ≼ Y ≼ Z and |X| = |Z|. Prove |Y|=|Z|.

$\qquad\qquad\qquad\qquad\qquad X \preceq Y \preceq Z$ and $|X| = |Z|$. Prove that $|Y|=|Z|$. Just started on cardinalities. Not sure about this one. Am I right if I do something along the lines ...
3
votes
4answers
131 views

If A is a denumerable set, and there exists a surjective function from A to B, then B is denumerable

I am having some trouble solving the following homework question and some help would be greatly appreciated!! Q: Prove that if $A$ is a denumerable set, and there exists a surjective function from ...
5
votes
1answer
124 views

Instance of Continuum Hypothesis implying cardinal inequality

I'm currently trying to solve Exercise 5.27 of Jech's Set Theory (3rd Millennium ed.), viz: If $2^{\aleph_1}=\aleph_2$, then $\aleph_{\omega}^{\aleph_0} \ne \aleph_{\omega_1}$. The presumption ...
2
votes
3answers
88 views

Cardinals of set operations without AC

Given info: $|A|=\mathfrak{c}$ , $|B|=\aleph_0$ in ZF (no axiom of choice). Prove: $|A\cup B|=\mathfrak{c}$ If $B \subset A\implies|A \backslash B|=\mathfrak{c}$? I have found several places ...
2
votes
1answer
78 views

Cardinality of a set containing subsets of $\omega_{1}$

Consider the set $ \{ X \subseteq \omega_{1} \ | \text{ such that } |X| = \aleph_{0} \} $ I know $\omega$ is in this set. But then I thought about it and realized that {2,3,4,... } was also in this ...
4
votes
5answers
1k views

Show that open segment $(a,b)$, close segment $[a,b]$ have the same cardinality as $\mathbb{R}$

a) Show that any open segment $(a,b)$ with $a<b$ has the same cardinality as $\mathbb{R}$. b) Show that any closed segment $[a,b]$ with $a<b$ has the same cardinality as $\mathbb{R}$. ...
0
votes
3answers
251 views

Proof of equal cardinality $|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$

How do I prove that the following sets have equal cardinality? $|\Bbb N \times\Bbb N \times\Bbb N| = |\Bbb N|$ ($|\Bbb N \times\Bbb N| = |\Bbb N|$ also for that matter) $|\Bbb Z \times\Bbb Z| = ...
0
votes
1answer
58 views

Deducing $|B^A|+|B^A|=|B^A|$ from $|A|+|A|=|A|$,

How attacking this question? Show that if $A$ and $B$ are sets such that $A$ is infinite, $|A|+|A|=|A|$, and $|B|\geq 2$, then $|B^A|+|B^A|=|B^A|$
0
votes
2answers
89 views

Cardinality calculation

How to simplify the following: $$2^{\aleph_0}(\aleph_0+\aleph_0)^{2^{\aleph_0}}$$ Thank you for every help.
5
votes
1answer
162 views

Exercise on partially ordered sets from Kunen's *Set Theory*

This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
0
votes
4answers
165 views

Comparing the cardinality of sets

An exercise is the following: Compare the cardinality of the following sets: The class of all real numbers $\mathbb{R} =: A$ The class of all polynomials $\mathbb{R}[X] =: B$ The class of all real ...
1
vote
2answers
303 views

Cardinality of Sets and Infinite Sets

The following are homework questions I would like assistance on. I will do what I can to work on these problems; any feedback is helpful. In the following problems, S is an infinite set (we do not ...
5
votes
2answers
812 views

How do you prove the trichotomy law for cardinal numbers?

Law of trichotomy is that for any two cardinals $a$ and $b$, exactly one of the conditions $a<b$, $a=b$, or $a>b$ holds.
4
votes
1answer
100 views

Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$

This is an exercise from Kunen's book. Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$ are equal. What I've tried: I want to prove by using induction on $m$. ...
0
votes
1answer
119 views

How to prove that $2^\omega=\mathfrak{c}$?

Let $\mathfrak{c}$ denote the continuum. My textbook says that $2^\omega=\mathfrak{c}$. How can one prove this equality? Thanks ahead:)
1
vote
1answer
103 views

Two questions about cardinals

I have two questions about set theory and cardinals. I know that $k_1, k_2, m$ are cardinals. I also know that $k_1 \leq k_2$. I need to prove that $(k_1)^m \leq (k_2)^m$. I understand from what ...
2
votes
1answer
198 views

How to prove that every infinite cardinal is equal to $\omega_\alpha$ for some $\alpha$?

How to prove that every infinite cardinal is equal to $\omega_\alpha$ for some $\alpha$ in Kunen's book, I 10.19? I will appreciate any help on this question. Thanks ahead.
3
votes
2answers
100 views

If a line segment is divided into 2 parts then one of the parts is equinumerous to the original segment.

Assume we have a set $X$ which is a (closed) line segment. Prove that if we split $X$ into 2 parts $X_1$ and $X_2$ then at least one of those sets would have the same cardinality as $X$. My attempt: ...
8
votes
2answers
435 views

axiom of choice: cardinality of general disjoint union

I have this exercise involving the axiom of choice, but I don't understand where it's needed: Let $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ be pairwise disjoint sets with $|X_i| = |Y_i|$. Prove, using ...
4
votes
4answers
1k views

Cardinality of the Irrationals [duplicate]

Possible Duplicate: Proof that the irrational numbers are uncountable We previously proved that $\mathbb{Q}$, the set of rational numbers, is countable and $\mathbb{R}$, the set of real ...
3
votes
2answers
196 views

How can I show that the set of reals and the set of pairs of reals have the same cardinality?

How can I show that the set of reals and the set of pairs of reals have the same cardinality? I know that since reals are uncountable infinite, I can't create a list of reals and talk about the ...
4
votes
2answers
1k views

Cartesian Product of Two Countable Sets is Countable

How can I prove that the Cartesian product of two countable sets is also countable?
4
votes
1answer
147 views

Prove that: $\aleph_0 \cdot \frak{c} = \frak{c} \cdot \frak{c}$

I've been fiddling with this enough. Found an answer here but didn't quite understand it. How do I prove that: $$\aleph_0 \cdot \frak{c} \leq \frak{c} \cdot \frak{c}$$ $$\frak{c} \cdot \frak{c} \leq ...
6
votes
2answers
287 views

prove cardinality rule $|A-B|=|B-A|\rightarrow|A|=|B|$

I need to prove this $|A-B|=|B-A|\rightarrow|A|=|B|$ I managed to come up with this: let $f:A-B\to B-A$ while $f$ is bijective. then define $g\colon A\to B$ as follows: $$g(x)=\begin{cases} ...
1
vote
2answers
244 views

Proof of cardinality inequality

I have this homework question I am struggling with: Let k1,k2,m1,m2 be cardinalities. prove that if $${{m}_{1}}\le {{m}_{2}},{{k}_{1}}\le {{k}_{2}}$$ then $${{k}_{1}}{{m}_{1}}\le {{k}_{2}}{{m}_{2}}$$ ...
3
votes
2answers
807 views

Cardinality of union of ${{\aleph }_{0}}$ disjoint sets of cardinality $\mathfrak{c}$

I have a home work question which is: " what is the cardinality of the union of ${{\aleph }_{0}}$ disjoint sets of cardinality $\mathfrak{c}$?" I believe somehow we can get to: cardinality = ...
12
votes
1answer
2k views

The cardinality of a countable union of countable sets, without the axiom of choice

One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have ...