# Tagged Questions

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 ...
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 ...
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. ...
48 views

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 ...
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 ...
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: ...
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 ...
74 views

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 $|B^A|+|B^A|=|B^A|$
89 views

### Cardinality calculation

How to simplify the following: $$2^{\aleph_0}(\aleph_0+\aleph_0)^{2^{\aleph_0}}$$ Thank you for every help.
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 $\text{c.c.}(\mathbb{P})$ ...
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 ...
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 ...
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.
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 $m$. ...
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:)
103 views

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 ...
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.
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: ...
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 ...
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 ...
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 ...
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?
147 views

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 $${{k}_{1}}{{m}_{1}}\le {{k}_{2}}{{m}_{2}}$$ ...
### 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 = ...
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 ...