Articles of elementary set theory

Does same cardinality imply a bijection?

This came up today when people showed that there is no linear transformation $\mathbb{R}^4\to \mathbb{R}^3$. However, we know that these sets have the same cardinality. I was under the impression that if two sets have the same cardinality then there exists a bijection between them. Is this true? Or is it just that any two […]

Where can I learn about the lattice of partitions?

A set $P \subseteq \mathcal{P}(X)$ is a partition of $X$ if and only if all of the following conditions hold: $\emptyset \notin P$ For all $x,y \in P$, if $x \neq y$ then $x \cap y = \emptyset$. $\bigcup P = X$ I have read many times that the partitions of a set form a […]

What does the completed graph of a function mean

zab said: the Levy metric between two distribution functions $F$ and $G$ is simply the Hausdorff distance $d_C$ between the closures of the completed graphs of $F$ and $G$. I have difficulty in understanding the sentence. In particular, what does the completed graph of a function $F$ mean? What are the two sets the Hausdorff […]

Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable

Since $X$ and $Y$ are countable, we have two bijections: (1) $f: \mathbb{N} \rightarrow X$ ; (2) $g: \mathbb{N} \rightarrow Y$. So to prove that $X\cup Y$ is countable, I figure I need to define some function, h: $\mathbb{N} \rightarrow X\cup Y$ Thus, I was wondering if I could claim something similar to the following: […]

$2^{\mathbb{N}}$ is uncountable — Power Set of Natural Numbers

Proof: Assume it is countable. Then we can arrange the sets in order (an enumeration) $A_1, A_2, A_3, … $. Now construct the set $ B = \{i \in \mathbb{N} \ | \ i \notin A_i \}$. Then $B=A_j$ for some $j$, a contradiction. This was the proof given in class and feels off. Can […]

Multiplication of Set Discrete math

I ran into some problems. Do you guys have any ideas about multiplication of set? For example: $A = \{1,2,3\},\quad B = \{x,y\},\quad C=\{0,1\}$ What will you do if I want to see: $A\times(B\times C)$ Can you do the binary method?

Can we conclude $\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa=2^{\aleph_\alpha}$ in ZFC?

In complex analysis, there is a function called Euler’s Gamma function. Whenever given a positive integer $n+1$, it will return $n!=\prod_{i=1}^{i < n+1}i$. I’m not sure if there is similar function for infinite cardinals such that $$\Gamma(\aleph_\alpha)=\prod_{\kappa \in Crd, \kappa=1}^{\kappa<\aleph_\alpha}\kappa$$, but at least we can evaluate the value of that production. So my question: Is […]

Symbol for the set of odd naturals?

Obviously the set of naturals is denoted $\mathbb{N}$, but is there a symbol for the set of odd naturals? Would $2\mathbb{N}+1$ (or $2\mathbb{N}-1$) be a standard notation?

Namesake of Cantor's diagonal argument

There are two results famously associated with Cantor’s celebrated diagonal argument. The first is the proof that the reals are uncountable. This clearly illustrates the namesake of the diagonal argument in this case. However, I am told that the proof of Cantor’s theorem also involves a diagonal argument. Given a set $S$, suppose there exists […]

Does continuity depend on the distance function?

I’m working through a book called “Introduction to Topology” and I’m currently working on a chapter regarding metric spaces and continuity. This is how my book defines continuity at a point: Let $(X,d)$ and $(Y,d’)$ be metric spaces, and let $a\in X$. A function $f: X\to Y$ is said to be continuous at the point […]