Articles of elementary set theory

Infinite set and proper subset.

Prove that a set A is infinite if and only if $A$ contains a proper subset $B$ that satisfies $|B|=|A|$. For the first part, I tried the following: Since $A$ is infinite, it has a countable subset $S=\{a_n\}$ with $n$ in $\mathbb N$. Then we have a function $f:A\to A\setminus \{a_0\}$ such that $f(a_n) = […]

Lebesgue integral on any open set is $\ge 0$, is it still $\geq 0$ on any $G_{\delta}$ set?

Definition of Lebesgue measurable function: Given a function $f: D \to \mathbb R \cup \{+\infty, -\infty\}$, defined on some domain $D \subset \mathbb{R}^n$, we say that $f$ is Lebesgue measurable if $D$ is measurable and if, for each $a\in[-\infty, +\infty]$, the set $\{x\in D \mid f(x) > a\}$ is measurable. If $f$ is an extended […]

Proof of non-existence of a continuous bijection between $\mathbb{R}$ and $\mathbb{R}^2$

There are a lot of websites and forums, which explain that there is a bijection between $\mathbb{R}$ and $\mathbb{R}^2$, and even give some bijections. (By the way: Can you generalize it? since it works with the natural Numbers and the real numbers, does there exist a bijection between any infinite set $X$ and $X^2$) On […]

Set convergence and lim inf and lim sup

I’m a bit confused with the general concept of convergence of a sequence of sets. I’m well aware that the limit of a sequence $\{C^{\nu}\}$ exists iff $$\liminf_{\nu \rightarrow \infty} C^{\nu} = \limsup_{\nu \rightarrow \infty} C^{\nu}$$ where lim inf (resp. lim sup) is the set of points that appear in the limit all but finitely […]

How to construct a function from a pair of possibly empty sets?

I am stuck on an elementary proof on the cardinality of sets on the following point: Given two possibly empty sets, $A$ and $B$, I need to prove the existence of any function $f:A\rightarrow B$. Is it possible? Perhaps using AC? I’m thinking you must have at least a non-empty $B$ (all elements of $A$ […]

Is there a set of all topological spaces?

This question is from Willard’s General Topology: Is there a set of all topological spaces? My try is: Suppose $\mathfrak T $ is set of all topological spaces, then $\mathfrak T $ ‘contains’ all the sets (i.e., if $S$ is some set, then $\{\varnothing, S\}\in\mathfrak T $). Since Willard assumes that a set cannot be […]

Maximum number of relations?

The question is that we have to prove that if $A$ has $m$ elements and $B$ has $n$ elements, then there are $2^{mn}$ different relations from A to B. Now I know that a relation $R$ from $A$ to $B$ is a subset of $A \times B$. And as the maximum number of subsets (Elements […]

If a set contain $(2n+1)$ elements and if the number of subsets which contain at most n elements is 4096, then what is the value of $n$?

If a set contain $(2n+1)$ elements and if the number of subsets which contain at most n elements is 4096, then what is the value of $n$?

Cardinality of subtraction of sets

Let $A_1$, $A_2$, $B_1$ and $B_2$ be sets such that $$ |A_1|=|A_2|,\quad|B_1|=|B_2|,\quad B_1\subset A_1,\quad B_2\subset A_2. $$ Where $|\ \ |$ denotes the cardinality of a set. How to show that $|A_1\setminus B_1|=|A_2\setminus B_2|$ ? We know that there is a bijection from $A_1$ to $A_2$ and one from $B_1$ to $B_2$, and we just […]

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 $n$.