Articles of elementary set theory

$A \subseteq B$ if and only if $B' \subseteq A'$?

hi already tried to prove it in my own way i would like to share my results hoping a mathematician somewhere could tell if I am right or wrong I think actually that the proof consists of 2 parts First part proof that $A \subseteq B$ implies $B’ \subseteq A’$ second part is the other […]

set theory proof (cartesian products/symmetric differences)

This question already has an answer here: elementary set theory (cartesian product and symmetric difference proof) 2 answers

How to prove $A=(A\setminus B)\cup (A\cap B)$

This question already has an answer here: Prove $A = (A \setminus B) \cup (A \cap B)$ 3 answers

Proof that the $(\mathcal P (\mathbb N),\triangle)$ is an abelian group?

This question already has an answer here: Does the Symmetric difference operator define a group on the powerset of a set? 3 answers

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!