Set Definition and 1000 Threads

Now, set of even integers is ## A = \{ \cdots, -4, -2, 0, 2, 4, \cdots \} ##. We need to prove that ## \mathbb{Z}^+ \thicksim A##. Which means that, we need to come up with a bijection from ##\mathbb{Z}^+## to ##A##. We know that ##\mathbb{Z}^+ = \{1,2,3,\cdots \} ##. I define the function ##f ...

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 believe that I am correct, the following statement here must be FALSE, right? It has to be false because A union B is like the two entire circles of the Venn diagram and that cannot be a subset of the intersection area, right? Now if this statement was flipped, then it would be true?

Hello, I am writing a blog post about Physics Constant. I recently found out how you can find out the Constant of Proportionality. and had an idea that all the Physics Constants were Constants of Proportionality. but I have no idea how to confirm this because there are so many. so I don't know...

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

Hey! :o let $A\neq \emptyset\neq B$ be sets, $C\subseteq A$, $D\subseteq B$ subsets and $f:A\rightarrow B$ a map. I want to show that the set $\{f^{-1}(\{x\})\mid x\in \text{im} f\}$ is a partition of $A$. To show that the set $\{f^{-1}(\{x\})\mid x\in \text{im} f\}$ is a partition of $A$, we...

I tried to find the element of best approximation ||t_0||≤||t||, ∀ y ∈ π Then |x_0|+|y_0|+|z_0| ≤|x|+|y|+|z| and we have x_0+2y_0+z=1 and x+2y+z=1. But I don't know hoe to continue...

Hey! :o I am looking at the following: There are the terms reflexive, symmetric, antisymmetric and transitive. Give for each combination of the properties (if possible) a set $M$ and a relation $R$ on $M$, such that $R$ satisfies these properties. What is meant exactly? Every possible...

Good morning, I'm trying to workout the time for an element to diffuse at set distance in microns. I have the distance, the diffusion coefficient, just unsure which equation I actually use. X= sqrt DT or the other one x= sqrt 2DT. I can't seem to figure out when you use one and not the...

Hey! :o Let $M:=\{7,4,0,3\}$. Determine $2^M$. Prove or disprove $2^{A\times B}=\{A'\times B'\mid A'\subseteq A, B'\subseteq B\}$. Let $a\neq b\in \mathbb{R}$ and $M:=2^{\{a,b\}}$. Determine $2^M$. Is there a set $M$, such that $2^M=\emptyset$ ? First of all how is $2^M$ defined? Is...

Homework Statement: See attached image. Homework Equations: ZFC set theory. Consider the text in the attached image. What is meant with "We require of an axiom system that it be possible to decide whether or not any given formula is an axiom."? Is consistency synonymous with soundness? Is...

Hi! So this is my first homework ever of Hamiltonian dynamics and I am struggling with the understanding of the most basic concepts. My lecturer is following Saletan's and Deriglazov's and from what I have read and from my lectures, this is what I think I know. Please let me know if this is...

The only restrictions are: -There are N points randomly distributed over an X-Y plane and it is necesary to pass every point at least 1 time, in order to get the path of the minimun distance throught all of them. -If does not matter how many times you pass any of this points. -You have to pass...

Dear Everyone I am having some difficulties on exercise 2e from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets A, B, C, and D. If a double implication fails, determine whether one or the other one of the possible...

Dear Every one, I am having some difficulties on exercise 2b and 2c from Topology 2nd ed by J. Munkres . Here are the directions: determine which of the following states are true for all sets $A$, $B$, $C$, and $D$. If a double implication fails, determine whether one or the other one of the...

In an elementary school version of set theory, we can take the complement of the empty set to obtain ##\emptyset ^ C = \mathbb{U}## However, in a sophisticated version of set theory, the concept of a "universal set" ##\mathbb{U}## is problematical. ( So says the current Wikipedia article on...

Hey guys, I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks: "Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal...

Gödel numbers are used to encode wffs of formal systems that are strong enough in order to deal with Arithmetic. In my question, Gödel numbers are used to encode wffs as follows: Syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such...

The above screen shot is from, @ the 1:43 mark. Which set of points is random? Answer below. The right set of points.

Hi all. I have another exam question that I am not so sure about. I've solved similar problems in textbooks but I have a feeling once again that the correct way to solve this problem is much simpler and eluding me. Especially because my answer to a) is already the solution to c) and d) (I did...

1. Let's show the three conditions for a subspace are satisfied: Since ##\mathbf{0}\in \mathbb{R}^n##, ##A\times \mathbf{0} = \mathbf{0}\in S##. Suppose ##x_1, x_2\in \mathbb{R}^n##, then ##A(x_1+x_2) = A(x_1)+A(x_2)\in S##. Suppose ##x\in S## and ##\lambda\in \mathbb{R}##, then ##A(\lambda x) =...

Let ##z = a + bi##. Using the definition of modulus, we have ##\vert z - 3 \vert < 2## is equivalent to ##\sqrt{(a+3)^2 + b^2} < 2##. Squaring both sides we get ##(a+3)^2 + b^2 < 4##. This is the open disk center at ##3## with radius ##4## which we write as ##D[-3, 2]##. First we show...

the text file of the data. 0 0 1 0.069478 2 0.137694 3 0.204332 4 0.269083 5 0.331656 6 0.391771 7 0.449167 8 0.503598 9 0.554835 10 0.602670 11 0.646912 12 0.687392 13 0.723961 14 0.756491...

Hello, I am having trouble with proving the following set identity: $$(A \cap B) \triangle C = (A \triangle C) \triangle (A \backslash B)$$ What I did so far was focus on this: $$((A \lor C) \land (\lnot A \lor \lnot C) \lor (A \land \lnot B)) \land ((\lnot A \land \lnot C) \lor (A \land C)...

I do not know what I'm doing wrong but I'm working on the problem of finding the normalization constants for the energy eigenstate equation for a 1D plane wave that is traveling from the left into a potential barrier where E < V at the barrier. This is from Allan Adams' Lecture 12 of his 2013...

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'm just having random thoughts today, and I didn't know where to put this, since this isn't even a homework problem. Anyway, is my way of writing the supremum of a set correct syntax-wise, or no?

Hi, a newbie to the site and hoping someone can help. Its been a long time since I studied physics or math at school. I am having some difficulty calculating the angular velocity of the arm trhough a partial movement (phase). The data set I have separates the angular velocity into the three...

For my base case I just used a graph with three vertices and 2 edges. Decomposing this would just give us the same graph, which has a path length of 2. The inductive step is where I'm having some trouble: One idea I have is that we take a graph G then inductively remove an edge to create two...

Determine if the number 7 is a natural number, an integer, a rational or irrational number. I know that integers include positive and negative numbers and 0. Let Z = the set of integers Z = {. . . -2, -1, 0, 1, 2, . . .} I also know that any integer Z can be written as Z/1 = Z. I will...

Determine if the set of vectors $\begin{bmatrix} x\\y\\5 \end{bmatrix}\in \Bbb{R}^3$ form a vector space ok if I follow the book example I think this is what is done $\begin{bmatrix} x_1\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5 \end{bmatrix} +\begin{bmatrix} x_2\\y_2\\5...

Goldrei's Propositional and Predicate Calculus states, in page 13: "The countable union of countable sets is countable (...) This result is needed to prove our major result, the completeness theorem in Chapter 5. It depends on a principle called the axiom of choice." In other words: the most...

I am given the following: $$u = (x,t)$$ $$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$$ and $$E = x + ct$$ $$n = x - ct$$ I need to solve for $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}{\partial t^2}$$ using the chain rule.How would I even...

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 the meaning...

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 Garling's...

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 formulate a proof of...

Hi Guys, The problem I am facing at the moment is to calculate the appropriate setpoint for a set of pumps supplying cold water to a building so that electricity and cost savings can be achieved. Here is the situation: There are three pumps connected in parallel. They are located on the ground...

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 to confirm my thinking on Proposition...

This is a pretty simple question, I am just trying to clear up confusion. Let ##D## be the rectangle in the plane with vertices ##(-1,0),(-1,1),(1,1),(1,0)##. Let ##\lambda >0##. Then what exactly does the set ##\lambda D## look like? Is it correct to say that, for example, ##2D## is the...

Summary: Potential at origin of an infinite set of point charges with charge (4^n)q and distance (3^n)a along x-axis where n starts at 1. From V=q/r, we find Vtotal=sum from 1 to infinity of (4/3)^n(q/a), which diverges. There cannot be infinite potential because there is a finite electric...

Has anybody ever heard of this? I learned about it in a discrete math class in grad school, and I've never heard of it anywhere else !? For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense: 0 OR 0 = 0. {0} UNION {0} = {0}. Similarly, logical AND & set...

Hi all, What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical...

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...

There is a statement on page 26 in Elliott Mendelson's book of "Introduction to Mathematical Logic" as shown: What I got from the statement above, which is obvious, I guess, is that in the sequence of \mathcal{B}_1,\mathcal{B}_2,...\mathcal{B}_k there are "SOME" well-formed formulas (wfs)...

In https://en.wikipedia.org/wiki/Recursively_enumerable_set, in the introductory section one reads "...a set S of natural numbers is called recursively enumerable... if: ...There is an algorithm that enumerates the members of S." and in a later section it says "According to the...

the solution set for the problem is x1=s+1 x2=t-2 x3=2s+2t x4=s-t+1I was thinking that I would have to isolate all the variables to one side and create a matrix and then get all the integers to another side and multiply them in order to get b but that doesn't seem correct to me. Can anyone...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... In Section 3.3 McInerney...

Hi. My question is regarding toy car A. If the car moves to the left, is the constant acceleration of ##2.40 \frac {cm} {s^2}## to the left or to the right?

Homework Statement (a) How many ways can at most three people out of a selection of ##n## applicants be selected for a job? (b) How many subsets of size at most three are there in a set of size ##n##? (c) How many ways can a given subset of size three or fewer be chosen for the job? Homework...

This is going to take a while to set up, so I apologize for that. This came up in the course of thinking about the Strong Law of Large Numbers. It's not homework. Suppose you have a doubly infinite sequence of random variables X_{i,n} that obey the following almost sure convergence relations...