Sets Definition and 1000 Threads

Mathematics uses Lebesgue measure for probability theory. However it is well known that it comes with a flaw that is not all sets are measurable. Is there a reason why the choice is also preferred in physics?

So the confusion here is that division by zero is often said to be undefined. So whereas, the point (0,0) certainly appears in the set of values where x=y, does the point (0,0) appear in the set of values where 1=y/x. Why or why not? In other words are the set of points where x=y the same as...

Was wondering if anyone knew of any good resources with programming challenges akin to the website "Project Euler" (about page found here). To be more specific, I'm looking for something consisting of problems centred around physics topics that would require some level of problem solving and...

I know how to solve each of those problems. For the set one, I look at the output of the S and try to match it with the input of T and then take the pair (input_of_S, output_of_T), and I do that for each pair. As for the formula one, I just plug in x = g(y). My confusion lies in trying to...

I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up. $$ S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \} $$ $$ S = \{ (i,j) : i...

Proof: Let ## p ## be an odd prime and ## G=\left \{ 1, 2, ..., p-1 \right \} ## be the set which can be expressed as the union of two nonempty subsets ## S ## and ## T ## such that ## S\neq T ##. Observe that ## p-1=22\implies p=23 ##. Let ## g\in G ##. Since ## g ## is either an element of ##...

So, my approach and solution are as follows: $$ [x * y] = \{ z \in M : z \sim (x * y) \} $$ Since we know that ##a * b \sim a^{\prime} * b^{\prime}## we can rewrite ##z## as ## x^{\prime} * y^{\prime} ##. Plugging this in yields: $$ [x * y] = \{ x^{\prime}, y^{\prime} \in M : x^{\prime} *...

The problem reads: ##f:M \rightarrow N##, and ##L \subseteq M## and ##P \subseteq N##. Then prove that ##L \subseteq f^{-1}(f(L))## and ##f(f^{-1}(P)) \subseteq P##. My co-students and I can't find a way to prove this. I hope, someone here will be able to help us out. It would be very...

I don't understand the solution: that for (1, ..., 1) the additive inverse is (-1, ..., -1), so the condition is not satisfied (and it is not a subspace). Which condition is not met? Thank you.

I am reading an abstract algebra textbook and enjoying it. I am working through preliminaries some more to refine my knowledge on proofs with sets before really digging in. I understand that if $$X \subseteq Y$$ and $$ Y \subseteq X$$ Then $$ X = Y$$ This makes sense to me. However, the...

I have been learning gr on YouTube for the last few months. Most of the videos and the book I have focus on high level understanding. I can do all of the tensor calculus proofs. However simple questions like how you set up a velocity vector or measure proper time in schwarzschild are beyond me...

Let ##S## be a set of n geometric objects in the plane. The intersection graph of ##S## is a graph on ##n## vertices that correspond to the objects in ##S##. Two vertices are connected by an edge if and only if the corresponding objects intersect. Show that the number of intersection graphs of...

Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?

A set is nothing more than a collection. To determine whether or not an object belongs to the set , we test it against one or more conditions. If it satisfies these conditions then it belongs to the set, otherwise it doesn't. The geometric point of view of sets- a set can be viewed as being...

Good evening! Have a look at the following part of a proof: Mentor note: Fixed the LaTeX I don't understand the use of implications. Isn't ##x\in C_M(A\cup B)\iff x\notin(A\cup B)##? To me, all of these predicates are equivalent.

Hey! :giggle: The below relations are given AUFTRAG($A_1, A_2, A_3$) and KUNDE($B_1, B_2, B_3$) with $A_1$ = AUFTRNR, $A_2$ = DATUM, $A_3$ = KUNDNR, $B_1$ = KUNDNR, $B_2$ = NAME and $B_3$ = ORT. Determine the below sets or justify why it is not possible to determine them. Let $A = A_1...

Hello! Reading a textbook I found that authors use the same trick to show that subsets of quotient topology are open. And I don't understand why this trick is valid. Below I provide there example for manifold (Mobius strip) where this trick was used Quote from "Differential Geometry and...

Ive read that Casual Set theory says that spacetime is made of spacetime atoms. But doesn't LQG says something similar? Or is just space atoms in LQG? A laypersons expansion fo the main difference would be very much appreciated? Also does Cause Set theory assume spacetime is fundamental rather...

Hello PF, After thinking, googling, and reading many threads here, I decided that I'm going to start learning calculus using Stewart. The problem is that the book puts too many exercises after each section, in addition to those at the end of each chapter. What's the difference between these...

This is for the lab report I have to submit. ##n## is the refractive index. ##L## is the length of a gas chamber and ##m## is the number of fringes passed as the pressure in the gas chamber changes by ##\Delta p##. We are already given the error in ##L##. I performed the experiment and obtained...

Can anyone please check/confirm my answers? I've shown my work and I boxed around all of my answers. Thank you.

Can anyone please check/confirm my answers if they are correct or not? I boxed around my answers just to be clear and understanding. Thank you.

I've always wondered why. Would the reason be that the air cools but the amount of moisture stays the same... which makes sweating less effective... which means no cooling effect even though the air is technically cooler.

Hey! :giggle: The set of $2$-dimensional orthogonal matrices is given by $$O(2, \mathbb{R})=\{a\in \mathbb{R}^{2\times 2}\mid a^ta=u_2\}$$ Show the following: (a) $O(2, \mathbb{R})=D\cup S$ and $D\cap S=\emptyset$. It holds that $D=\{d_{\alpha}\mid \alpha\in \mathbb{R}\}$ and...

I'm looking to pre-calculus physics textbooks with problems sets. I need something as rigorous as possible. Problem sets can be separate from textbooks

I got my last post locked due to insulting a member here. I do apologize I'm under a lot of stress and was nothing personal. The post was about the level of difficulty of engineering courses compared to those of mathematics at the undergraduate level. I am just curious if anyone would be kind...

It says that sets that are equal ie having the same types of elements can also be equivalent, having the same number of elements when b has more distinct elements than a. Please explain

Suppose ##f## is not uniformly-continuous. Then there is ##\epsilon>0## such that for any ##\delta>0##, there is ##x,y\in K## such that if ##|x-y|<\delta##, ##|f(x)-f(y)|\geq \epsilon##. Choose ##\delta=1##. Then there is a pair of real numbers which we will denote as ##x_1,y_1## such that if...

Definition: A set is sequentially compact if all sequences contained in the set contain a subsequence that converges to a point in the set. Let ##N\in \mathbb{N}## and suppose that ##m\geq N##. Let ##x\in K_m##. Since ##K_m\subset K_{m-1}\subset \ldots \subset K_N##, it follows that ##x## is an...

Define a collection of open sets to be denoted as ##P_i##, ##1\leq i\leq N## where ##N\in \mathbb{Z}^+##. Let ##x\in\cap_{i=1}^N P_i##. By definition, ##x## must belong to every single ##P_i##. In particular, ##x\in P_1## and ##x\in P_2##. Since ##P_1## and ##P_2## are open, there exist...

I have a question that is related to categories and physics. I was reading this paper by John Baez in which he describes a TQFT as a functor from the category nCob (n-dimensional cobordisms) to Vector spaces. https://arxiv.org/pdf/quant-ph/0404040.pdf. At the beginning of the paper @john baez...

Upper bound definition for sets: $ M \in \mathbb{R} $ is an upper bound of set $ A $ if $ \forall \alpha\in A. \alpha \leq M$ Upper bound definition for sequences: $ M \in \mathbb{R} $ is an upper bound of sequence $ (a_n)$ if $ \forall n \in \mathbb{N}. a_n \leq M$ Suppose we look at the...

Hey! 😊 Let $v\in \mathbb{R}^2$ be a vector and let $X\subseteq \mathbb{R}^2$ be a subset, then we define the subset of $\mathbb{R}^2$ : $$v+X:=\{v+x\mid x\in X\}$$ Let \begin{equation*}L:=\mathbb{R}\begin{pmatrix}1 \\ 2\end{pmatrix}=\left \{\begin{pmatrix}x \\ y \end{pmatrix}\mid 2x-y=0\right...

Summary:: Be the set X of vectors {x1,...,xn} belong to the vector space E. If this set X is convex, prove that all the convex combination of X yet belong to X. Where convex combination are the expression t1*x1 + t2*x2 + ... + tn*xn where t1,...,tn >= 0 and t1 + ... + tn = 1 I tried to suppose...

By ZFC, the minimal set satisfying the requirements of the axiom of infinity, is the intersection of all inductive sets. In case that the axiom of infinity is expressed as ∃I (Ø ∈ I ∧ ∀x (x ∈ I ⇒ x ⋃ {x} ∈ I)) the intersection of all inductive sets (let's call it K) is defined as set K = {x...

Question: How many elements are in each set? For the first set, I think it's 8995 because the set is the union of {1,2,3,4,5},{1,2,3,4,5,6},...{1,2,3,...9000}. So 9000 - 5 = 8995. For the second set, I'm not too sure about counting the elements in the set. Since 1<x≤i, I can't think of any x...

Hi, I am looking for a solution that generates combinations of objects from a series of objects in a set. For example, {Apple, Pear, Orange} should bring back Apple Pear Orange Apple, Pear Apple, Pear, Orange, Apple, Orange ... Items in the series should not repeat (i.e. Apple, Orange /...

Hello there.We know that we have sets of numbers like the real numbers, complex numbers, quaternions, octonions.Could we find a set of numbers more general than that of real numbers that has basic properties of the real numbers like commutativity, order, addition, multiplication, division and...

I have a hypothesis of which I wonder if it's sound. Perhaps you guys can advise me: Suppose ##x_n\Rightarrow a_n## (logical implication) for some set X and set A. I think we have to assume a bijection. Then, if ##a_m = False##, ##x_m## should be ##False##, right? So, in case of a bijection...

Homework Statement:: x Relevant Equations:: x I stumbled upon the following example in the book - " How to prove it, A structured approach " ( 2nd edition) , Vellerman. Homework Statement:: He then asks to describe the set: ## \bigcup_{s \in S} L_{s} \, \backslash \, \bigcup_{s \in S}...

From Zorich, Mathematical Analysis II, sec. 11.5.2: where as one can read from the statement, the sets could also be unbounded. I do not report here the proof of the fact a), beacuse I have no doubt about it and one can, without the presence of dark steps in the reasoning, assume a) as...

I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help in order to fully understand the set of Borel sets ... ... The relevant text reads as follows: My questions related to the above text are as follows:QUESTION 1...

I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ... I need help in order to fully understand the set of Borel sets ... ... The relevant text reads as follows: My questions related to the above text are as follows:QUESTION...

I am looking for a formula that will them me how many possible connections between 2 sets of objects. Ex: 12 on left and 4 on right, how many possible states can it be in? state one: top on left connected to top on right state two: top on left connected to 2 on right state three: top on left...

Let x ∈ A1 ∪ A2 then x ∈ A1 or x ∈ A2 If x ∈ A1, as A1 is open, there exists an r > 0 such that B(x,r) ⊂ A1⊂ A1 ∪ A2 and thus B(x,r) is an open set. Therefore A1 ∪ A2 is an open set. How does this prove that A1 ∪ A2 is an open set. It just proved that A1 ∪ A2 contains an open set; not that...

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focused on Searcoid's treatment of ZFC in Chapter 1: Sets ... I need help in order to fully understand the Axiom of Power Sets and Definition 1.1.1 ... The...

For the past 8 years I have used weight sets like the one below and they have been very versatile. But, just before Christmas, most of them were stolen. The person was nice enough to leave me some of the 100g and below masses. I've finally gotten permission to get new sets and now I can't find...

Proof: Let ##A, B \in \mathcal{O}## and ##x \in A \cap B##. Then there exists ##\varepsilon_A, \varepsilon_B > 0## such that ##B_{\varepsilon_A}(x) \subset A## and ##B_{\varepsilon_B}(x) \subset B##. Let ##\varepsilon = \min\lbrace\varepsilon_A, \varepsilon_B\rbrace##. Then ##B_\varepsilon(x)...

In a discrete metric space open balls are either singleton sets or the whole space ... Is the situation the same for open sets or can there be sets of two, three ... elements ... ? If there can be two, three ... elements ... how would we prove that they exist ... ? Essentially, given the...

I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 1: Set Theory and Metric Spaces and am currently focused on Section 2: Metric Spaces ... ... I need help in order to fully understand Example 2.7(a) ... .. The relevant text reads as follows:My...