Articles of elementary set theory

What is meant by 'ordering of set by inclusion'?

I was recently introduced to partial order relation. Though I could understand the relation, I couldn’t grab the sense of the above quoted term used for the ordering of sequence(??) of subset. Can anyone help me clarify the concept with a intuitive example?

All Sets have bijection with cartesian products of Subsets?

I was doodling around with some math today, trying to find “representations” for sets as cartesian products of their proper subsets. For example: $\mathbb{N}\leftrightarrow 2\mathbb{N}\times\{0,1\}$ $\mathbb{Z}\leftrightarrow 2\mathbb{Z}\times\{0,1\}$ $\mathbb{R}\leftrightarrow \mathbb{Z}\times[0,1)$ I’m thinking most ways to do this have to do with abstract algebra, using quotient structures times the respective substructure. But I was wondering given an […]

Is the Cartesian product of an infinite number of $\mathbb{Z}^+$ countable?

We know the Cartesian product of a finite number of $\mathbb{Z}^+$ is countable, for example, $\mathbb{Z}^+ \times \mathbb{Z}^+ \times \mathbb{Z}^+$, because, let $m, n, p$ be from each of the $\mathbb{Z}^+$, we can find a one-to-one function $2^m 3^n 5^p$ that maps them to a subset of $\mathbb{Z}^+$. But what if we increase the number […]

Construct a bijection $F(C, A \times B)\to F(C ,A) \times F(C, B)$

Let $A, B, C$ be sets. Construct a bijective function $F(C, A \times B)\to F(C ,A) \times F(C, B)$. Here, $F(C, A \times B)$ is the set of functions $f: C \to A \times B$. Could someone show me the way how to do this? What questions should I ask myself? I understand that $F(C, […]

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?