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)...

25 views

Question about dictionary orders for $(\mathbb{Z}_+^{\omega})$

I just want to make sure I understand the explanation stated below. According to the order relation stated below we have $(a_0,a_1,...) < (b_0,b_1,..)$ if $a_i = b_i$ for finitely many values ...
43 views

Subtle difference between statemetns invloving negation in set theory.

What is the difference between the statements $x$ is not in an infinite number of sets $E_n$ and ...
45 views

Relations, Ordered Pairs, Naive set theory by Halmos

I quote: "Explicitly: a set R is a relation if each element of R is an ordered pair;" The question is: "what about the converse? is a set of ordered pairs could be considered a relation?"
28 views

25 views

Prove that if $R$ is a symmetric, transitive relation on $A$ and the domain of $R$ is $A$, then $R$ is reflexive on $A$.

Assume, $R$ is a symmetric, transitive relation on $A$ and the domain of $R$ is $A$. $Dom(R)=A$ implies $(\forall x \in A)(\exists y \in A)[xRy]$. Since, $xRy$ is true it follows that $yRx$ is ...
96 views

How are sets “detached” from their structure?

This question is best asked with an example. Consider the real numbers. However we construct the real numbers, the "final product" so to speak, is not just a set, but it is a complete ordered field. ...
42 views

Describe the equivalence classes generated by T

Suppose $S = \{(x,y) \in \mathbb{R}^2\mid y = x + 1\text{ and } 0 < x < 2\}$. Question Describe the equivalence relation T on the real line that is the intersection of all equivalence ...
49 views

29 views

Commonplace sets

I recently started reading about sets of numbers, set builder notation, and operations on sets of numbers. To practice using different symbols (e.g., $\wedge$) and different set "layouts," I decided ...
45 views

Understanding notation for the sequence definition

Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old? Definition. A sequence ($s_n$) is said to diverge ...
34 views

Cardinality of subsets with finite intersections

Let $\ F_0$ be a family of disjoint subsets of $C$. $\ |C|= \aleph_0$. Prove that $\ (*) |F_0|\leq\aleph_0$. This part was relatively simple, in the presence of choice an injection can be ...
16 views

Chartrand Mathematical Proofs 3e Exercise 1.45

I'm self-studying this book to learn how to do proofs, I have previously studied Calculus 1,2,3 and Linear Algebra in college in the US. I have a problem with the following question: Exercise. 1.45 ...
34 views

Existence of an inverse relation for $R \subseteq A \times A$.

I'm stuck with the following problem: Given the set $A = \{1,2,3,4,5\}$, construct a relation $R \subseteq A \times A$ such  R \circ R^{-1} = \triangle_A = \{(a,a) \hspace{5pt} | \hspace{5pt} a \...
28 views

Proving that $A+B - (A \cap B) = A \cup B$ for binary integers

I hope computing questions are fine here. I'm trying to show that for all binary numbers $A$ and $B$, $A+B - (A \cap B) = A \cup B$. It's confusing me firstly because I'm not sure what the "set ...
34 views

For each of the following sets, determine its cardinality (ω, 2ω, or something else) and prove that your answer is correct

(a) A1 = {f ∈ (ω → ω) : ∀n,m ∈ ω (n < m ⇒ f(n) < f(m))}. (b) A2 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω f(m) ≤ n}. (c) A3 = {f ∈ (ω → ω) : ∃n ∈ ω∀m ∈ ω (n ≤ m ⇒ f(n) = f(m))}. a) A1 = {f ∈ (ω → ω) : ∀n,...
95 views

Let $\mathscr{A}$ be a set of sets. Let's denote $\{A \setminus B : A,B \in \mathscr{A}\}$ by $\mathscr{A} \setminus \mathscr{A}$. The Marica-Schönheim theorem in combinatorics says that $|\... 1answer 38 views Assistance with finding the accumulation points for$(3,6) \cup (6,9]$I'm having trouble digesting the definition of an accumulation point(s). Can you help me to understand it given the following:$(3,6) \cup (6,9]$I know this produces the interior set$(3;9]\...
Let $\ A\subset R$ have the following characteristic: For all $\ a,b \in A$ , $\ \frac{a+b}{2} \notin A$. Prove that there exists a maximal set A. Prove its cardinality is $\ \aleph$. The first ...
My set theory notes state that the following is a 'bad' definition for the cardinality of a set $x:$ $|x|=\{y:y\approx x\}$ \$(y\approx x\ \text{iff} \ \exists\ \text{a bijection}\ f:x\rightarrow y )...