Articles of elementary set theory

Distributivity of ordinal arithmetic

Let greek letters be ordinals. I want to prove $\alpha(\beta + \gamma) = \alpha\beta + \alpha\gamma$ by induction on $\gamma$ and I already know it holds true for $\gamma = \emptyset$ and $\gamma$ a successor ordinal. Let $\gamma$ be a limit ordinal. I found $$ \alpha(\beta + \gamma) = \alpha \cdot \sup_{\epsilon < \gamma} (\beta […]

How to derive a union of sets as a disjoint union?

$$\bigcup_{n=1}^\infty A_n = \bigcup_{n=1}^\infty (A_{1}^c \cap\cdots\cap A_{n-1}^c \cap A_n)$$ The results is obvious enough, but how to prove this

Elementary Set Theory Question

I am working on the following question. Let $X$ be a nonempty set and consider a map $f:X\to Y$. Prove that the following are equivalent: (a) $f$ is injective; (b) there exists $g:Y\to X$ such that $g∘f=1_{X}$ where $1_{X}:X\to X$ is the identity map; (c) for any set $Z$ and any maps $h_{1},h_{2}:Z\to X$, the […]

Can a collection of subsets of $\mathbb{N}$ such that no one set contains another be uncountable?

Let C be a collection of subsets of $\mathbb{N}$ such that $A,B\in C \Rightarrow A \not\subseteq B$. Can C be uncountable?

Recursive Mapping

I was wondering about what is the general definition of a recursive mapping between any two sets. Is there a condition for a mapping to be able to be written in recursive form? Is the following claim true: A mapping can be represented recursively if and only if its domain is a set that can […]

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 […]