Articles of elementary set theory

Universality of $(\mathbb Q,<)$

I have few questions to the proof on Universality of linearly ordered sets. Could anyone advise please? Thank you. Lemma: Suppose $(A,<_{1})$ is a linearly ordered set and $(B,<_{2})$ is a dense linearly ordered set without end points. Assume $F\subseteq A$ and $E\subseteq B$ are finite and $h:F \to E$ is an isomorphism from $(F,<_{1})$ […]

Injective Equivalence

I’m trying to prove that these two statements are equivalent. I’ve already proven that $f$ injective implies that $$f^{-1} \left(f(B)\right) = B$$ but I need to show that $$f^{-1} \left(f(B)\right) = B \Leftrightarrow f\left(\bigcap A_t\right)=\bigcap f\left(A_t\right). $$ Any advice would be greatly appreciated!

finding sup and inf of $\{\frac{n+1}{n}, n\in \mathbb{N}\}$

Please just don’t present a proof, see my reasoning below I need to find the sup and inf of this set: $$A = \{\frac{n+1}{n}, n\in \mathbb{N}\} = \{2, \frac{3}{2}, \frac{4}{3}, \frac{5}{4}, \cdots\}$$ Well, we can see that: $$\frac{n+1}{n} = 1+\frac{1}{n} > 1$$ Therefore, $1$ is a lower bound for $A$, however I still need to […]

Simple Question on 1-1, Increasing integer functions

So my brain is frazzled which is probably why this seems like a big deal to me right now, but I just can’t get over this reasoning: Suppose you have $$F = \{\text{all } 1\text{-}1, \text{increasing functions } \Bbb N \to \Bbb N\}$$ $1$-$1$: means that every value of the domain maps to some unique […]

Epic-monic factorisation in $\mathbf{Set}$.

I’m stuck on Exercise 5.2.1 of Goldblatt’s “Topoi: A Categorial Analysis of Logic“: Given a function $f:A\to B$, if $h\circ g: A\twoheadrightarrow C\rightarrowtail B$ and $h’\circ g’: A\twoheadrightarrow C’\rightarrowtail B$ are two different epic-monic factorisations of $f$ (i.e. $f=h\circ g=h’\circ g’$), then there exists a unique $k:C\to C’$ such that commutes, and furthermore $k$ is […]

Without AC is there a relationship between $\beth$ and $\aleph$ numbers?

Assuming AC we know that all $\beth_\alpha$’s will be $\aleph_\beta$ for some $\beta$ since they can be well ordered. Can anything interesting be said about their relationship without AC? Is it possibly consistent that with the exception of $\beth_0$ none of the $\beth$’s can be well-ordered?

To prove the elementary statement without using AC and AF

I have little knowledge in set theory and I have difficulty in proving the following statement without using Axiom of Choice or Axiom of Foundation: Let $A$ be a set. Then there exists a set $B$ satisfying the following conditions: $A\cap B=0$ and there exists a one-to one function $f$ from $A$ onto $B$. I […]

Book suggestion on set theory/logic

Can anyone recommend good books/tutorials on set theory/logic with simple explanations for a person with no math background (nothing beyond arithmetic and basic algebra back in school)?

Show that $\left(\bigcup_{i \in I}A_{i}\right )^{c}=\bigcap_{i \in I}A_{i}^{c}$

I’m learning for a test and I think there will be some task similar to this one. I’d like to know if I did it correct and if not can you please say how to do it correctly? Let $\Omega$ be a set, let $I$ be an index-set and let $A \subseteq \Omega$ and $A_{i} […]

Definition of Substitution (Equality vs Equivalence relations)

I’m confused as to wether substitution is a consequence of the properties about equivalence relations (transitivity, reflexivity, and symetry) or rather it is an independent property of equality. For example, in general $(x=y) \wedge (x<z) \Longrightarrow (y<z)$. But what if I have an equivalence relation , say $”\equiv$”, instead of equality. Can I use substitution […]