Articles of elementary set theory

Semi-partition or pre-partition

For a given space $X$ the partition is usually defined as a collection of sets $E_i$ such that $E_i\cap E_j = \emptyset$ for $j\neq i$ and $X = \bigcup\limits_i E_i$. Does anybody met the name for a collection of sets $F_i$ such that $F_i\cap F_j = \emptyset$ for $j\neq i$ but $X = \overline{\bigcup\limits_i F_i}$ […]

What is bigger, $p(\mathbb{N})$ or $\mathbb{R}$?

This question already has an answer here: Is $2^{|\mathbb{N}|} = |\mathbb{R}|$? 2 answers

Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered. So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it possible that exist an infinite set $A$ with hartog number $h(A)=\aleph_0$?

Image of a union of collection of sets as union of the images

I am having problems with establishing the following basic result. Actually, I found a previous post that is close in nature (it is about inverse image), but I was interested in this specific one, with the following notation, because it is the one I found in the book I am self-studying and I guess that […]

Set Addition vs. Set Union

Given two sets $A$ and $B$, what is the difference between $A + B$ and $A \cup B$? For example, if $A = \left\{ a, b, c \right\}$ and $B = \left\{ d, e, f \right\}$, what are $A + B$ and $A \cup B$, respectively?

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 […]