Suppose we have a finite collection of sets $A_1$,…,$A_n$. Is there an algorithm which gives a new collection $B_1$,…,$B_m$, which consist of pairwise disjoint sets, $\cup B_i=\cup A_j$ and each $B_i$ is a subset of some $A_j$?

Recall that for $a,b$ an element of the set of real numbers and $a<b$, the closed interval $[a,b]$ in real numbers is defined by $[a,b]=\{x$ is an element of reals|$a\leq x\leq b\}$. Show that the given intervals have the same cardinality by giving a formula for a one to one function $f$ mapping the first […]

Prove the following statement by mathematical induction: For all integer $n \ge 2$, $n^3-n$ is divisible by 6 My attempt: [Proof] Let the given sentence p(n) (1) $2^3-2$=6 is divisible by 6. p(2) is true. (2) Suppose for all integer $k \ge 2$, p(k) is true. That is, mathematical hypothesis is $k^3-k$ is divisible by […]

I am reading Naive Set Theory. On Section 3 (Unordered Pairs), page 11, it is written that: As further examples, we note that $$\{x:x\neq x\} = \varnothing$$ and $$\{x:x= a\} = \{a\}.$$ In case $S(x)$ is $(x \in’ x)$, or in case $S(x)$ is $(x=x)$, the specified $x$’s do not constitute a set. The last […]

I am reading some notes on correspondences and have a question. (The notes are here.) I have a question about something on page 1. Basically, the notes provide some motivation for why we might want to define correspondences. It then says, We would like to have a notion of a set-valued function. The seemingly obvious […]

I am trying to learn about proofs and one of the exercice in my book (Maths ABC) is about proof by contradiction. I think I understand the concept but I would like to have a feedback on the following proof about whether or not it is correct. Let $A$ be the following proposition: $$[(S\cup P […]

I often hear of identification $\mathbb{R^2} \cong \mathbb{C}$. Exactly what kind of isomorphism is there? Are we considering groups? Fields? Topological spaces? Or is it even a strict equality? For example, can we say that the Heine-Borel theorem about compact sets in $\mathbb{R^n}$ holds for $\mathbb{C}$, meaning that the identification mentionned above is a homeomorphism […]

We need to show that there is a bijection $h \colon \mathbb{Z}_+ \times \mathbb{Z}_+ \to \mathbb{Z}_+$. For this purpose, Munkres define a subset $A$ of $\mathbb{Z}_+ \times \mathbb{Z}_+$ as follows: $$A \colon= \{ \ (x,y) \in \mathbb{Z}_+ \times \mathbb{Z}_+ \ \colon \ y \leq x \ \}.$$ Then he defines the maps $f \colon \mathbb{Z}_+ […]

This question already has an answer here: Prove $f(S \cup T) = f(S) \cup f(T)$ 3 answers

Let $(\Omega,\mathcal{A})$ be a measurable space. $\mathcal{N}\subset\mathcal{P}(\Omega)$ is called a $\sigma$ ideal, if $$ (1)~\emptyset\in\mathcal{N},~~~~~(2) N\in\mathcal{N}, M\subset N\Rightarrow M\in\mathcal{N},~~~~~(3)(N_n)\in\mathcal{N}^{\mathbb{N}}\Rightarrow\bigcup_n N_n\in\mathcal{N} $$ Show that for every $\sigma$-ideal $\mathcal{N}$ it is $$ \sigma(\mathcal{A}\cup\mathcal{N})=\left\{A\Delta N|A\in\mathcal{A},N\in\mathcal{N}\right\}. $$ Hint: $$ \left\{A\Delta N|A\in\mathcal{A},N\in\mathcal{N}\right\}=\left\{B\subset\Omega|\exists A\in\mathcal{A},N\in\mathcal{N}: B\setminus N=A\setminus N\right\} $$ I do not have a special idea, to be honest. For the inclusion […]

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