# Tagged Questions

This tag is for elementary questions on set theory, spanning topics usually found in introductory courses in set theory, in addition to review sections of graduate textbooks in the same field. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, (un)...

45 views

28 views

### Given $\mathcal{F}$ a filter on $X$, and either $A$ or its complement are in $\mathcal{F}$, then $\mathcal{F}$ is ultrafilter

I am trying to prove the following fact: Given $\mathcal{F}$ a filter on $X$, Suppose $\forall A \subseteq X$ where either $A$ or $X\backslash A$ are in $\mathcal{F}$, then $\mathcal{F}$ is ...
57 views

42 views

### Inf and Supremum of $\{\arctan(x) \; : \; x \in \mathbb R\}$

I had trouble with this, but I think it's because there is no minimum and no maximum, correct? For the $\inf$ I got $-\frac{\pi}{2}$ and for the $\sup$ I got $\frac{\pi}{2}$ But on second thought I ...
42 views

### Let $E = \{2k$| $k\in \mathbb{N}\}$ ; $F = \{l^2$| $l\in \mathbb{N}\}$. Determine $E\cap F$, $E-F$. [closed]

I have attempted to solve these questions on my own but as I have not seen an example of this before I have no idea how to present the answer or if I am even on the right track. Any help or some type ...
35 views

### Verify sup, max, min, inf for $\{\frac{n+3}{n}\;:\;n \in \mathbb N\}$

I think I'm finally starting to understand this and wanted to check that I answered the following correctly: $\{\frac{n+3}{n}\;:\;n \in \mathbb N\}$ inf: 1 min: 4 sup: DNE max: DNE
13 views

27 views

### Determine Intersection and difference of infinitely countable sets

I've been confused about how exactly write the answer to this question if you could help me out I would greatly appreciate it. Let $A = \{3^k$| $k\in \mathbb{N}\}$ ; $B = \{l^3$| $l\in \mathbb{N}\}$. ...
44 views

45 views

### Consider the set $S=\{-1,0, 1\}$, what is $A=\{xy: x, y\in S\}$?

Consider the set $S=\{-1,0, 1\}$ Set $A=\{xy: x, y\in S\}.$ Find all elements of $A$. Is it $A=\{-1,0\}$ or $A=\{-1,0, 1\}?$ Can you multiply $1$ by itself? Because it's $x$ and $y$....
19 views

### Proving $g(x) = E_x$.

I'm pretty new to mathematical proof and set theory and I'm having trouble proving this problem. I've started on it, but I don't know where to progress. Problem: Suppose that $f: A \rightarrow B$. ...
27 views

### If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$

Claim: If $A = \coprod_{n \in \mathbb{N}} A_n$ is uncountable, then there exists an uncountable $A_n$ $\coprod$ is the disjoint union of disjoint sets $A_n \subset A, \forall n \in \mathbb{N}$ Is ...
42 views

### Cardinality of the Domain vs Codomain in Surjective (non-injective) & Injective (non-surjective) functions

I'm a student in college just beginning to study the basics of set theory. In studying about Surjective & Injective functions & how they map their domain to their codomain, it came to my mind ...
47 views

34 views

### What is the result of a natural number power the cardinality of an infinite set?

What is the result of a natural number power the cardinality of an infinite set? Is it the cardinality of the infinite set? Thank you!
44 views

### Compact Sets of $(X,d)$ with discrete metric

Let $X \neq \emptyset$. Define the discrete metric on $X$ with: $d(x,y)=\left\{\begin{array}{ll} 1, & x \neq y \\ 0, & x=y\end{array}\right.$ (a) Ascertain the compact ...
35 views

### Cardinality of an infinite set divided by the cardinality of another infinite set (or itself) [duplicate]

Is the cardinality of an infinite set divided by the cardinality of another infinite set indeterminate? And what if it is divided by itself? Have these results been proven or are they unprovable? ...