Articles of elementary set theory

Are there an infinite set of sets that only have one element in common with each other?

In a card game called Dobble, there are 55 cards, each containing 8 symbols. For each group of two cards, there is only one symbol in common. (The goal of the game being to spot it faster than the other players, which is not the point of my question). If I translate that to mathematical […]

Proving Dedekind finite implies finite assuming countable choice

I’d like to show that if a set $X$ is Dedekind finite then is is finite if we assume $(AC)_{\aleph_0}$. As set $X$ is called Dedekind finite if the following equivalent conditions are satisfied: (a) there is no injection $\omega \hookrightarrow X$ (b) every injection $X \to X$ is also a surjection. Countable choice $(AC)_{\aleph_0}$ […]

Does there exist any uncountable group , every proper subgroup of which is countable?

Does there exist an uncountable group , every proper subgroup of which is countable ?

How to prove that $\mathbb{Q}$ ( the rationals) is a countable set

I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it equivalent to find a bijection from $\mathbb{Q}$ to $\mathbb{Z}$?

What are good books/other readings for elementary set theory?

I am looking to expand my knowledge on set theory (which is pretty poor right now — basic understanding of sets, power sets, and different (infinite) cardinalities). Are there any books that come to your mind that would be useful for an undergrad math student who hasn’t taken a set theory course yet? Thanks a […]

The set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?

How can I prove that the set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable? Edit: This answer came to mind. Is it correct? This answer just came to mind. By contradiction suppose the set is $\{f_n\}_{n \in \mathbb{N}}$. Define the function $f: \mathbb{N} \to \{0,1\}$ by $f(n) \ne f_n(n)$. Then $f \notin\{f_n\}_{n […]

de morgan law $A\setminus (B \cap C) = (A\setminus B) \cup (A\setminus C) $

First part : I want to prove the following De Morgan’s law : ref.(dfeuer) $A\setminus (B \cap C) = (A\setminus B) \cup (A\setminus C) $ Second part: Prove that $(A\setminus B) \cup (A\setminus C) = A\setminus (B \cap C) $ Proof: Let $y\in (A\setminus B) \cup (A\setminus C)$ $(A\setminus B) \cup (A\setminus C) = (y […]

Two questions about equivalence relations

Question 1: Let $x,y \in S$ such that $x\sim y$ if $x^2 =y^2\pmod6 $. Show that $\sim$ is an equivalence relation. This is what I tried: Reflexive: $x^2\pmod6 = x^2$ implying $x\sim x$ Symmetry: suppose $x\sim y$, then $x^2 =y^2\pmod6$ Operating by inverse of $x^2$ both sides we have $e=(x^2)^{-1} y^2\pmod6$ Then I have $y^2=x^2\pmod6$, […]

What is the set-theoretic definition of a function?

I’m reading through Asaf Karagila’s answer to the question What is the Axiom of Choice and Axiom of Determinacy, and while reading the explanation of Bertrand Russell’s analogy (“The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.”) at the bottom, I […]

Is a contradiction enough to prove a set equality to $\varnothing$?

An exercise asks me to prove the following: $$A \cap (B-A) = \varnothing$$ This is what I did: I need to prove that $A \cap (B-A) \subseteq \varnothing$ and $\varnothing \subseteq A \cap (B-A)$ The second one seems to be obvious by definition (not sure if it is fine to say that in a test). […]