Sets Definition and 1000 Threads

I tried saying z = x + iy, then squared both sides so that I would get something that looked like: |z - i|^2 + |z + i|^2 + |z - i||z + i| = 3, where the first two terms are simple but the third term is what I don't know what to do with. I'm wondering if I'm using the wrong approach. For that...

I need a little help with Baby Rudin material regarding the convergence of a sequence of sets please. I wish to follow up on this thread with a definition of convergence of a sequence of sets from Baby Rudin (Principles of Mathematical Analysis, 3rd ed., Rudin) pgs. 304-305: (pg. 304)...

I tried to name the shaded area of a Venn diagram using numbers to isolate the regions. And I found that there are several ways to get the same region. Can the set notations simplfy

I'm trying to recreate this document in LaTeX, but I'm not sure how they aligned "Factor" and "Solution" after 1.1.1. Any ideas?

Is this related to pigeon principle? $$S_1=\{1,2,3,4\},$$ $$S_2=\{2,3,4,5\},$$ $$S_3=\{4,5,6,7\},$$ $$S_4=\{5,6,7,8\},$$ $$S_5=\{7,8,9,10\},$$ $$S_6=\{8,9,10,11\},$$ $$S_7=\{5,6,2,4\},$$ $$S_8=\{1,5,7,9\},$$ $$S_9=\{4,8,10,11\},$$ $$S_{10}=\{5,7,10,11\}$$ When we choose two of them, there is...

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of Theorem 3.1.16 Stoll's statement of Theorem 3.1.16 and its proof reads as follows: Can someone please help me to demonstrate a formal and rigorous proof of the following:If $$U = X \cap O$$ for...

Zermelo-Fraenkel Axioms - the Axiom of Choice (ZFC), is conceptually incoherent. To me, they stole Cantor’s brilliant work and minimized it. Replies?

The uncountable sets [0,1] and [0,2] have the same cardinality ##2^{\aleph_0}##. Yet the second set is twice as big as the first set, in the sense of measure theory. Is there something similar for countable sets, by which we can say that the set of integers is twice as big as the set of odd...

Hey! :o Let $A,B$ be sets, such that $A\times B=B\times A$. I want to show that one of the following statements hold: $A=B$ $\emptyset \in \{A,B\}$ I have done the following: Let $A$ and $B$ be non-empty set. Let $a\in A$. For each $x\in B$ we have that $(a,x)\in A\times B$. Since...

Hey! :o We have the subsets \begin{equation*}V:=\left \{\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}\mid x_1=0\right \}, \ \ \ W:=\left \{\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix}\mid x_2=2\right \}, \ \ \ S:=\left \{\lambda \begin{pmatrix}1 \\ 0 \\ -1\end{pmatrix}\mid \lambda \in...

Hey! :o We have the following subsets: \begin{align*}&U_1:=\left \{\begin{pmatrix}x \\ y\end{pmatrix} \mid x^2+y^2\leq 4\right \} \subseteq \mathbb{R}^2\\ &U_2:=\left \{\begin{pmatrix}2a \\ -a\end{pmatrix} \mid a\in \mathbb{R}\right \} \subseteq \mathbb{R}^2 \\ &U_3:=\left \{\begin{pmatrix}x...

I am trying to learn some topology and was looking at a problem in the back of the book asking to show that a topological space with the property that all set are closed is a discrete space which, as understand it, means that all possible subsets are in the topology and since all subsets are...

{1, 2 ,3} = {1, 2, 3, 3, III}? {1, 2 ,3} = {one, dos, three}? {Tom, Dick, Harry} = {Thomas, Richard, Harrison}? Seems to me, these are undetermined until the set's "type" or "category" definition of its members is defined so as to determine what elements are members of the set... whether...

The interval ##[0,1]## of real numbers has a non-zero measure. The set of all rational numbers in the interval ##[0,1]## has zero measure. But there are also sets that are somewhere in between, in the sense that their measure is neither zero nor non-zero. They are sets for which measure is not...

1. I consider this problem algebraically, ##c\cdot \vec{u}+(1-c)\cdot \vec{v}=c(1,2)+(1-c)(2,1)=(c,2c)+(2-2c,1-c)=(2-c,1+c)##; since the constraint I know is ##c\geq 0##, I can conclude the expected vectors##(x,y)## must have ##x\leq2, y\geq 1##. 2. Similarly, I get...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows: In the above proof by...

For the set A: Note that if n is odd, then ## A = \{ -1 + \frac {2} {n} : \text{n is an odd integer} \} ## . If n is even, A = ## \{1 + ~ \frac {2} {n} : \text{ n is an even integer} \} ## . By a previous exercise, we know that ## \frac {1} {n} ## -> 0. Let ## A_1 ## be the sequence when n...

The Definition of a Neighborhood and the Definition of an Open Set ... Carothers, Chapters 3 & 4 ... I am reading N. L. Carothers' book: "Real Analysis". ... ... I am focused on Chapter 3: Metrics and Norms and Chapter 4: Open Sets and Closed Sets ... ... I need help with an aspect of...

Problem Statement: 40 lbs Relevant Equations: 10 times 4 I wanted to know if anyone has moved 40lbs using 4 sets of 10lbs motors instead of a motor that can move 40lbs I am testing using 4 small motors instead of 1 big motor thank you

Dear all, I have two small questions regarding operations on sets. (1) Prove that \[A\subseteq B\subseteq C\] if and only if \[A\cup B=B\cap C\]. (2) What can you say about sets A and B if \[A\B = B\] ? In the case of (1), I have used a Venn diagram and I understand why it is true, but...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am reading Chapter 6: Topology ... ... and am currently focused on Section 6.1 Topological Spaces ... I need some help in order to fully understand a statement by Browder in Section 6.1 ... ... The...

I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented. Now, I'm having a hard time to grasp the idea/motivation behind the...

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 4: Topology of R and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.3.4 ... ... Theorem 4.3.4 and its proof read as follows: In the above proof by...

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with the proof of Lemma 1.2.11 ... Duistermaat and Kolk"s Lemma 1.2.11 reads as follows: Can someone please demonstrate...

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Lemma 1.2.5 ... Duistermaat and Kolk"s Lemma 1.2.5 reads as follows:In the above proof by Duistermaat and Kolk...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help in order to understand some...

My question concerns the portion of the proof stating, “...we set up a correspondence between the elements of U(A_n), for n in N, and a subset of S by making the element a correspond to (m, n) if A_m is the first set in which a appears, and a is the nth element of A_m.” In particular, I am...

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.3: Open and Closed Sets ... Conway's Example 5.3,4 (b) reads as follows ... ... Note that Conway defines open and closed...

I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 5: Metric and Euclidean Spaces ... and in particular I am focused on Section 5.3: Open and Closed Sets ... Conway's Example 5.3,4 (b) reads as follows ... ... Note that Conway defines open and closed sets...

My Question : 1.Why are the inequalities considered? Why not simply use ##n(A\cap B) = n(A)+ n(B)-n(A\cup B)## to get ## n(A\cap B) = 39## ? 2. The way I interpret this is : If the set for people liking cheese was to be a subset of the set for people who like apples then the most number of...

Homework Statement Prove or disprove: if A⊆B∪C, then A⊆B or A⊆C. Homework EquationsThe Attempt at a Solution I am unsure of how to go about proving this. I know that A is a subset of B union C then A is a subset of B or A is a subset of C and I understand what a subset is and what a union is...

Homework Statement Let A= {1, 2, 3}, B= ℤ+, C= [1, infinity) That is C= {x∈ℝ:x≥1} What is B - A and C - A? Homework EquationsThe Attempt at a Solution I am unsure of how to go about answering this. I know that B - A means what elements are in B that aren't in A. Would that make the answer...

Consider a set $A$ and its subsets $B$ and $C$. It is known that $|A-(B\cap C)|=8$, $|B|=5$, $|C-B|=1$ and $|B\cap C|=3$ (here $-$ denotes set difference). How many subsets $X\subseteq A$ are there if $X\cap B\cap C\ne\emptyset$, $|X-(B\cup C)|\ge3$ and $|X\cap (B-C)|=2$?

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...

I identified what appears to be a partitioning of all integers > 1 into mutually disjoint sets. Each set consists of an infinite series of integers that are all the powers of what I am calling a "root" r (r is an integer that has no integer roots of its own, meaning: there is no number x^n that...

Q1: Write all proper subsets of S = {1, 2, 3, 4 }. Q2: Let S = {1,2,5,6 } Define a relation R on S of at least four order pairs, as (a,b)  R iff a*b is even (i.e. a multiply by b is even)...

Homework Statement Show that it is not necessarily true that the infinite union of closed sets is closed Homework EquationsThe Attempt at a Solution From intuition, I came up with the following counter-example: ##\displaystyle \bigcup_{n=2}^{\infty} \left[ \frac{1}{n}, \frac{n}{n+1} \right] =...

Homework Statement I'm having issues understanding a mistake that I'm making, any assistance is appreciated! I know a counterexample but my attempt at proving the proposition is what's troubling me. Prove or disprove $$P(A \cup B) \subseteq P(A) \cup P(B) $$ Homework EquationsThe Attempt at...

Homework Statement Suppose ##R## and ##S## are relations on a set ##A##. If ##R## and ##S## are transitive, is ##R \cup S## transitive? Why? Homework EquationsThe Attempt at a Solution Suppose that ##a## is an arbitrarily but particularly picked element of ##R \cup S##, then $$a \in R \...

Homework Statement Prove: If A and B each have at least two elements, then not every element of P(A×B) has the form A1 ×B1 for some A1 ∈ P(A)and B1 ∈ P(B). Homework EquationsThe Attempt at a Solution Suppose A = {1, 2}, B = {3, 4}. AXB = {(1,3), (1,4), (2,3), (2,4)} P(A) = {{1}, {2}, {1,2}...

I am working on a proof problem and I would love to know if my proof goes through: If $A, B$ are sets and if $A \subseteq B$, prove that $|A| \le |B|$. Proof: (a) By definition of subset or equal, if $x \in A$ then $x \in B$. However the converse statement if $x \in B$ then $x \in A$ is not...

Hey! :o I want to describe in words the following sets: 1. $A:=\{(x,y)\in \mathbb{R}^2\mid x>0, y\leq 1\}$ $A$ is the set of all pointgs where the first coordinate is positiv and the second one is less or equal to $1$. It is the subarea of the plane that is under the point $(0/1)$ to...

It is well known that the set of exponential functions ##f:\mathbb{R}\rightarrow \mathbb{R}_+ : f(x)=e^{-kx}##, with ##k\in\mathbb{R}## is linearly independent. So is the set of sine functions ##f:\mathbb{R}\rightarrow [-1,1]: f(x) = \sin kx##, with ##k\in\mathbb{R}_+##. What about...

Hey! :o Show that the set of all positive rational numbers is a countable set. (Hint: Consider all points in the first quadrant of the plane of which the coordinates x and y are integers.) Show that the union of a countable number of countable sets is a countable set. I have done the...

Hello! (Wave) I want to find the supremum, infimum of the following sets: $$\{ x \in \mathbb{R}: 0<x^2-1<3\}, \{1+\frac{(-1)^n}{n}: n=1,2, \dots \}$$ For the first set I have thought the following: $$ 0<x^2-1<3 \Rightarrow 1<x^2<4 \Rightarrow x^2>1 \text{ and } x^2 <4 \Rightarrow (x>1 \text{...

Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...

Just wanted to know if the work is sound and logical on my paper posted above. I realized I probably should have included notation for the power of the sets. This is my first attempt at theorem proving in Introductory Real Analysis. I realize now that I’m starting into a subject that...

I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.6 Constructions on Categories ... I need some further help in order to fully understand some further aspects of Example 1.8, Page 17 ... ... Example 1.8, Page 17 ... reads as follows:I find the...

-Definition of complete space: if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in converges in M. (and from this definition we can define Hilbert Space) -Definition of Hilbert space: A Hilbert space is a vector space with an...

why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?