Articles of elementary set theory

Construction of bijection between set of functions

$f : X \leftrightarrow Y$ and i want to construct bijection between $\lbrace g : X \rightarrow A \rbrace$ and $\lbrace g : Y \rightarrow A \rbrace$. How to do this?

Preimage of Intersection of Two Sets = Intersection of Preimage of Each Set : $f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$

This question already has an answer here: how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$ 2 answers

Question about a proof about singular cardinals

The following is a lemma in Just/Weese on page 179: Lemma 17: Let $\kappa$ be an infinite cardinal. Then $\kappa$ is singular iff there exist an $\alpha < \kappa$ and a set of cardinals $\{\kappa_\xi : \xi < \alpha \}$ such that $\kappa_\xi < \kappa$ for every $\xi < \alpha$ and $\kappa = \sum_{\xi < […]

Proving equality of sets

How do I prove the following equations (I am new to statistics and not sure where to begin even after trying to figure it out): (a) $A – B = A – A \cap B = A \cup B – B$ (b) $A \mathbin{\Delta} B = A \cup B – A \cap B$

Are there countably many infinities?

$\aleph_0$, $\aleph_1, \aleph_2$ and so on are indexed by a natural number so shouldn’t there be countably many infinities?

Prove that the set of integers bounded below is well-ordered.

Given that we have some $r \in \mathbb{R}$, demonstrate that the set $ S=\{n\in \mathbb{Z} : r\leq n\}$ is well-ordered. I first tried to demonstrate that there exists exactly one integer on the half-open interval $(r,r+1]$ (because I had used that as a Lemma for a previous assignment) and then proceed inductively to show that […]

Finite family of infinite sets / A.C.

Let $\{A_i\mid i\in n\}$ be a finite family of infinite sets. ( That is, $A_i$ is infinite for every $i\in n$ and $n\in \mathbb{N}$) Here, we can choose representative $a_i$ from each $A_i$ and construct $\{a_i\mid i\in n\}$. This process really doesn’t use Axiom of Choice? How do I write this process down formally (in […]

The norm of a $d$-tuple of operators in Hilbert space

In this question $\mathcal{H}$ stands for a Hilbert space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, with inner product $\langle\cdot\;| \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators from $\mathcal{H}$ to $\mathcal{H}$. Let ${\bf T} = (T_1,…,T_d) \in \mathcal{B}(\mathcal{H})^d$ we define the norm of ${\bf T}$ by \begin{eqnarray*} \|{\bf T}\| &=&\sup\left\{\bigg(\displaystyle\sum_{k=1}^d\|T_kx\|^2\bigg)^{\frac{1}{2}},\;x\in […]

Equivalence relation on $\mathbb{N}$ with infinitely many equivalence classes

Does there exist an equivalence relation, defined on $\mathbb{N}$, with infinitely many equivalence classes, all of which contain infinitely many elements? I see no reason for such a relation not to exist, but I’m having trouble finding an example.

countable-uncountable intuition behind a set

let $Q$ := $\{q_1, …,q_n,…\}$ be an enumeration of the rationals. let $\epsilon_n$ = $\frac{1}{2^n}$ and let $S_i$ = $(q_i-\epsilon_i, q_i + \epsilon_i)$ Let $S$ be the union of all $S_i$. Take the complement of $S$ in $\mathbb{R}$. I have a few queries about this set. I’m sure they are all somewhat trivial but i’ve […]