Articles of elementary set theory

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 […]

∅ ⊆ { ∅ } Is this true or false?

True or false? Im guessing true because an empty set is a subset of every set. Is this a correct assumption? The only time an empty set is not a subset of something is when its a proper subset of an empty set correct?

Why is $\emptyset$ a subset of every set?

Please read the details of this question before answering. I am reading a book in which they say $\emptyset \subseteq S$ for any set $S$. One justification they give for this is that if $\emptyset \not \subseteq S$, then there must be some element in $\emptyset$ that is not in $S$. And we conclude that […]

Cofinality of $2^{\aleph_\omega}$

Is the following statement correct: $\operatorname{cf}(2^{\aleph_\omega})=\aleph_0$? It appears in the “Jech” book. Wikipedia however states that $\operatorname{cf}(\aleph_\omega)=\aleph_0$. The exact statement in the “Jech book” is on page 165. It is as follows: “Thus $2^{\aleph_0}$ cannot be $\aleph_\omega$, since cofinality($2^{\aleph_{\omega}})=\aleph_0$.”

inclusion of sets -transitive?

show that if $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$ Can I do it with using injective functions? $A\subseteq B$ means there exists an injective fcn $f:A\to B$ $B\subseteq C$ means there exists an injective fcn $g:B\to C$ then the composition $g\circ f:A\to C$ is also an injective function then $A\subseteq C$ in each […]

How can $\mathbb{R} \subset \mathbb{C}$?

My book defines $\mathbb{C}$ as $\{(x, y) : x,y \in \mathbb{R} \}$. So even when the imaginary part is zero, the elements of $\mathbb{C}$ are ordered pairs, while the reals are not. I’m not sure if I should make a different question for this, but my book says that the reals are embedded in $\mathbb{C}$. […]

Proof of complement of intersection defined using an arbitrary set.

I hate asking questions like these (ones where I have no idea what it is talking about). Let $ \xi$ be a collection of sets and define $$I= \bigcap \{ F|F \in \xi \}\quad\text{and}\quad U= \bigcup \{ F|F \in \xi \}.$$ Prove that $I^{c}= \bigcup \{ F^{c}|F \in \xi \}$ and $U= \bigcap \{ F^{c}|F […]