Articles of elementary set theory

Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$.

Let $A,B$ and $U$ be sets so that $A\subseteq U$ and $B\subseteq U$. Prove that $A\subseteq B$ iff $(U\setminus B)\subseteq(U\setminus A)$. The forward implication is easy to prove but i got stuck at the backward implication, i.e. $(U\setminus B)\subseteq(U\setminus A)\Rightarrow A\subseteq B$. Can someone help me out with it? Thank you.

If A is infinite, does there have to exist a subset of A that is equivalent to A?

I was reading Rudin and there is an alternative definition of infinite set (the first definition is “not finite”): A is infinite if it is equivalent to one of its subset. Then I was wondering the converse: if A is infinite, does there have to exist a subset of A that is equivalent to A?

What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to?

What is the set of all functions from $\{0, 1\}$ to $\mathbb{N}$ equinumerous to? I have figured out the question when it’s the other way around, but I am not making any progress here. The worst thing is, I don’t know how to think of such problems, and even when it was functions from $\mathbb{N}$ […]

(Verification) $C \approxeq A$ or $C \approxeq \text{section of } A$ $\forall C$ s.t. $C \subset A$

Claim Let A is well-ordered set. $C \approxeq A$ or $C \approxeq \text{section of } A$ $\forall C$ s.t. $C \subset A$ Proof If $C=A$, $C \approxeq A$ since a well-ordered set is order-isomorphic to oneself. If $C \neq A$, there $\exists a \in A$ s.t. $a \notin C$. Then $A\setminus C = \{a \in […]

Cross product of the reals question

Is $\Bbb {R} \times \Bbb {R} \subseteq \Bbb {R}$? If this is the case then would it be true that $|\Bbb {R} \times \Bbb {R}| \leq |\Bbb {R}|$?

Proving the Cantor Pairing Function Bijective

How would you prove the Cantor Pairing Function bijective? I only know how to prove a bijection by showing (1) If $f(x) = f(y)$, then $x=y$ and (2) There exists an $x$ such that $f(x) = y$ How would you show that for a function like the Cantor pairing function?

Non-existence of a Surjective Function from a Set to Its Subsets (Cantor's theorem)

Show that: Let A be a set and let $P(A)$ be the set of all subsets of $A$. Then there is no surjection $f: A→P(A)$. Here is what I thought: if $A=\{a,b\}$ then it has only two elements where $P(A)=\{∅,\{a\},\{b\},\{a,b\}\}$ has 4 elements. Therefore $f:A→P(A)$ cannot be surjective. But I have some problems: 1) How […]

Intuition behind Cantor-Bernstein-Schröder

The book I am working from (Introduction to Set Theory, Hrbacek & Jech) gives a proof of this result, which I can follow as a chain of implications, but which does not make natural, intuitive sense to me. At the end of the proof, I found myself quite confused, and had to look carefully at […]

Prove that the family of open sets in $\mathbb{R}$ has cardinality equal to $2^{\aleph_0}$

Let $\mathcal{T}$ be the family of all open sets in $\mathbb{R}$. Show that $| \mathcal{T}|=2^{\aleph_0}$ $\textbf{My Attempt:}$ I know that $\forall A \in \mathcal{T}$. $A$ is the countable union of open intervals with rational end points. I want to use the Cantor-Bernstein Theorem. That is I need to find injective functions $f$ and $g$ such […]

Sequences of sets property

I’m having trouble to prove the following question: Supose $\{A_n\}_{n\in\mathbb{N}}$ is a family of sets such that $A_1\subset A_2\subset A_3\subset\ldots$ (it’s possible to have $A_n=A_{n+1}$). I need to prove that $$\lim_{n\to\infty}A_n = A_1\cup\bigcup_{n=2}^\infty(A_{n}\backslash A_{n-1}).$$ I’m not sure if this is useful, but I proved that $A_{n}\backslash A_{n-1}$ and $A_{n’}\backslash A_{n’-1}$ are disjoint if $n\neq n’$. […]