Articles of elementary set theory

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 […]

Improving my understanding of Cantor's Diagonal Argument

I studied Cantor’s Diagonal Argument in school years ago and it’s always bothered me (as I’m sure it does many others). In my head I have two counter-arguments to Cantor’s Diagonal Argument. I’m not a mathy person, so obviously, these must have explanations that I have not yet grasped. My first issue is that Cantor’s […]

Why is “for all $x\in\varnothing$, $P(x)$” true, but “there exists $x\in\varnothing$ such that $P(x)$” false?

This question already has an answer here: Why is predicate “all” as in all(SET) true if the SET is empty? 6 answers

“The set of all true statements of first order logic”

In one of my lectures, the lecturer put a bunch of examples of sets on the board, stuff like the set of all humans, set of all well typed programs in some programming language, the set of all true statements of first order logic, the set of all proofs of them, and the set of […]

$\cap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$

$A$ and $B$ are sets and $\mathcal{F}$ is a family of sets. I’m trying to prove that $\bigcap_{A \in \mathcal{F}}(B \cup A) \subseteq B \cup (\cap \mathcal{F})$ I start with “Let $x$ be arbitrary and let $x \in \bigcap_{A \in \mathcal{F}}(B \cup A)$, which means that $\forall C \in \mathcal{F}(x \in B \cup C)$. So, […]

Is any relation which contains only one ordered pair transitive?

I need clarification. Let $A=\{1,2,3\}$ be a set and $R=\{(1,2)\}$ be a relation on $A$. Is it a Transitive relation? I am confused because some text books say $R$ is transitive if it contains only one ordered pair. I am not able to explain why $R$ can said to be transitive in the above case. […]

Is $1$ a subset of $\{1\}$

Is the number $1$ a subset of the set $\{1\}$ just as $\{1\}$ is a subset of the set $\{\{1\}\}$? I’m a little bit confused because $1$ is an element not a set…

Is every subset of $\mathbb{Z^+}$ countable?

This question already has an answer here: An infinite subset of a countable set is countable 2 answers