Articles of elementary set theory

What is the negation of this statement?

Let $(K_n)$ be a sequence of sets. What is the negation of the following statement? For all $U$ open containing $x$, $U \cap K_n \neq \emptyset$ for all but finitely many $n$.

Is it always true that $\sum^{\infty}a_{i}1_{A_{i}}-\sum^{\infty}b_{i}1_{B_{i}}=\sum^{\infty}c_{i}1_{C_{i}}$?

Suppose $\sum^{\infty}a_{i}1_{A_{i}}\geq \sum^{\infty}b_{i}1_{B_{i}}$, where $a_{i},b_{i}\geq 0$ and the sets possibly intersect i.e. $A_{i}\cap A_{j}\neq \varnothing $ and same with $B_{i}$. Is it true that we can write $\sum^{\infty}a_{i}1_{A_{i}}-\sum^{\infty}b_{i}1_{B_{i}}=\sum^{\infty}c_{i}1_{C_{i}}$ for $c_{i}\geq 0$? If we have disjointness i.e. $A_{i}\cap A_{j}=\varnothing $ and same for $B_{i}$ then yes. My concern is that when I proved it the general […]

Build a bijection $\mathbb{R} \to \mathbb{R}\setminus \mathbb{N}$

This question already has an answer here: Find a bijection from $\mathbb R$ to $\mathbb R-\mathbb N$ 6 answers

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do 'length' and 'size' differ?

I read that ordinal numbers relate to length, while cardinal numbers relate to size. How do ‘length’ and ‘size’ differ? Note : I am an absolute novice, and I’m having a little trouble visualizing ordinal numbers.

Prove a function is one-to-one and onto

I need some help proving the following function is one-to-one and onto for $\mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$. $F(i, j) = {i + j – 1 \choose 2} + j$ I know you guys like to see some attempt at a problem but I honestly have no idea where to start. A naive attempt simply […]

If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

If the empty set is a subset of every set, why it isn’t written with the elements of a set? like so $\{1,2,3,\emptyset\}$ Or why isn’t $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements and the other has 1 but since the empty set is a subset of both, then why it isn’t being mentioned explicitly […]

$A$ uncountable thus $\mu(A)>0$

I was thinking if it is possible to come up with a $\sigma$-finite measure on $\mathbb R$ which is positive on any uncountable set. I think that I have a proof that there is no such measure – but I am not sure if it is formal enough. The proof: let $\mu$ be such measure, […]

Name for collection of sets whose intersection is empty but where sets are not necessarily pairwise disjoint

According to Wolfram MathWorld, a collection of sets $A_1, A_2, \ldots, A_n$ is said to be disjoint if $A_i \cap A_j = \emptyset$ for all $i \ne j$. In other words, ‘disjoint’ refers only to ‘pairwise disjoint’. I am looking for a name for a collection of sets where $A_1 \cap A_2 \ldots \cap A_n […]

Formal definition for indexed family of sets

Essentially I’d like to know the formal definition of the object $\{A_{i}|i\in I\}$ .This is my context: 1.- From Wikipedia (Here) I understand that a family of elements in $S$ indexed by $I$ and denoted by $\{A_{i}|i\in I\}$ is a function $A:I\longrightarrow S$. 2.- From Hrbacek’s book (Introduction to set theory): “We say that $A$ […]

What are the bounds (upper and lower) for $|A+A|$?

Let $A$ be a finite set of real (or complex) numbers. If I consider sets with small sizes, we have that: If $A$ is the empty set, then $A+A$ is also empty. If $A$ is a singleton, then $A+A$ is also a singleton. If $|A|=2,$ then $|A+A|=3.$ If $|A|=3,$ then $|A+A|$ can be at most […]