Articles of elementary set theory

Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.

I tried doing this by defining a function $f$ which takes an element (a subset of $\mathbb{N}$) and maps it to an infinite sequence of $0$s & $1$s. A subset, i.e the element of $P(\mathbb{N})$ will contain discrete numbers from $\mathbb{N}$ (arranged in increasing order, since the order of numbers in a subset does not […]

Proving that Zorn's Lemma implies the axiom of choice

I am trying to solve an exercize, in which we were asked to prove that Zorn’s lemma implies axiom of choic. I am using a guidness that was given that said we should use a set $\mathcal{F}$ which i’ll define throughout the proof: Proof: Given a set of nonempty sets $F$, define the set $\mathcal{F}$ […]

How many different subsets of a $10$-element set are there where the subsets have at most $9$ elements?

How many different subsets of a $10$-element set are there where the subsets have at most than $9$ elements? I know there are $2^{10}$ total number of $10$-element sets. Please explain to me this problem.

Maximal and Minimal Elements

My textbook gives an example for finding maximal and minimal elements on a set. The set is $(\{2,4,5,10,12,20,25\},|)$. It says to draw a Hasse diagram to find the maximal and minimal elements of the set, saying that the elements on the “top” of the diagram are the maxima, and the ones on the bottom are […]

How can set-builder notation be used to create a set from a function?

My understanding of the set-builder notation (from this question), is that the format for defining a set $C$ is as follows: $ C ::= $ { $x \in S: \varphi(x)$} read as “C is the set of all $x$ in $S$ such that $\varphi(x)$ is true.“ From this explanation, $x$ is iterated over the elements […]

Can anyone clarify how a diverging sequence can have cluster points?

$p$ is a cluster point of $S\subset M$ if each neighborhood of $p$ contains infinitely many points. Here is my confusion, a cluster point is also a limit point of $S$, right? If so, then how does the sequence $((-1)^n)$, ${n\in \mathbb N}$ has two cluster points namely $1, -1$ especially since the sequence does […]

Proving $(0,1) $ is not countable

Recall that a countable set $S$ implies that there exists a bijection $\mathbb{N} \to S.$ Now, I consider $(0,1).$ I want to prove by contradiction that $(0,1)$ is not countable. First, I assume the contrary that there exists a bijection $f,$ and I can find an element in $S,$ but not in the range of […]

The union of a sequence of countable sets is countable.

While working on the theorem below, I constructed the following proof: Theorem. If $\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$ is a sequence of countable sets, then $$ \bigcup_{n\in\mathbb N}E_{n} $$ is countable. Proof. Let $S=\left\langle E_{n}\right\rangle_{n\in\mathbb{N}}$ be a sequence of countable sets. Moreover, let $x_{n,m}$ be the $m$th element of the $n$th set in $S$. Construct a sequence of […]

How to prove $f(\bigcap_{\alpha \in A}U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$?

$f(\bigcap_{\alpha \in A} U_{\alpha}) \subseteq \bigcap_{\alpha \in A}f(U_{\alpha})$ Suppose $y \in f(\bigcap_{\alpha \in A} U_{\alpha})$ $\implies f^{-1}(y) \in \bigcap_{\alpha \in A} U_{\alpha} \implies f^{-1}(y) \in U_{\alpha}$ for all $\alpha \in A$ $\implies y \in f (U_{\alpha})$ for all $\alpha \in A \implies y \in \bigcap_{\alpha \in A}f (U_{\alpha})$ $\bigcap_{\alpha \in A}f(U_{\alpha}) \subseteq f(\bigcap_{\alpha \in A} […]

Inductive definition of power set for finite sets

I’m stuck on a problem using recursive definitions: Let $X$ be a finite set. Give a recursive definition of the set of all subsets of $X$. Use Union as the operator in the definition. I can see how the union of all subsets separately gives a set of all subsets, but I don’t understand how […]