Sets Definition and 1000 Threads

Homework Statement Give combinatorial proofs of the identities below. Use the following structure for each proof. First, define an appropriate set S. Next, show that the left side of the equation counts the number of elements in S. Then show that, from another perspective, the right side of the...

Hey all, the schema theorem shows that in all probability a genetic algorithm will converge to a solution. much like the second law of thermodynamics for optimization. Although, it is taught with the genes being $$ \in (0,1, *), * \in (0,1) $$ is there a proof for non binary genes? example...

Let A be an open set and A=(a,b). Can A be described, as closed set as "or every x>0, all the elements of closed set [a+x,b-x] are elements of A"?

Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution So I have to show that each side is a subset of the other side Assume x∈ A ∪ (∩Bi) so x∈A or x∈∩Bi case 1 x∈ ∩ Bi so x∈ (B1∩B2∩B3...∩Bn) which implies x∈B1 and x∈B2 ... and x∈Bn so x∈B1∪A and x∈B2∪A...

Homework Statement Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n. Homework Equations S_n = Symmetric set ≅ = isomorphism Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...

Homework Statement For each ##n \in \mathbb{N}##, let ##A_{n}=\left\{n\right\}##. What are ##\bigcup_{n\in\mathbb{N}}A_{n}## and ##\bigcap_{n\in\mathbb{N}}A_{n}##. Homework Equations The Attempt at a Solution I know that this involves natural numbers some how, I am just confused on a...

Came to know about the following problem from a friend which can be simplified to the following: A1, A2, ...Am and B1, B2,...Bn are two groups of sets each group spanning the sample space. Now there are p elements in each of Ai and each element is in exactly p1 of the sets of the A group. Again...

If the Euclidean plane is partitioned into convex sets each of area A in such a way that each contains exactly one vertex of a unit square lattice and this vertex is in its interior, is it true that A must be at least 1/2? If not what is the greatest lower bound for A? The analogous greatest...

Hi. Can we infer something about physics from stuff like Vitali sets or the Banach-Tarski paradox? Maybe if we assume the energy in a given space volume to be well defined and finite, that there must be fundamental particles that can't be split, or that there must be a Planck length and energy...

I'm given the example that the space $\mathbb{R}^2$ is spanned by each of the following set of vectors: $$\left\{i, j\right\}$$, $$\left\{i, j, i+j\right\}$$, and $$\left\{0, i, -i, -j, i+j\right\}$$. However, it's not obvious to me how. Let $i = (s, t)$ and $j= (u, v)$ then $\left\{i...

Hi, I can derive a few properties of the limit inferior and limit superior of a sequence of sets but I have trouble in understanding what they actually mean. However, my understand of lim inf and lim sup of a sequence isn't all that bad. Is there a way to understand them intuitively (something...

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

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

How would I solve this using the algebra of sets?

Suppose that A\B is disjoint from C and x∈ A . Prove that if x ∈ C then x ∈ B . So I know that A\B∩C = ∅ which means A\B and C don't share any elements. But I don't necessarily understand how to prove this. I heard I could use a contrapositive to solve it, but how do I set it up. Which is P...

First of all sorry for my english skills. 1. Homework Statement Im trying to get the set levels of this function: f(x,y)=(x-y)/(1+x^2+y^2)=z Homework Equations circle-> (x-xo)^2+(y-yo)^2=r^2 The Attempt at a Solution (Leaving this here just to give a graph...

Hi So I am learning about sets and I wanted to know if these definitions was correct, specifically the properties of sets under operations, and I had a question. please help. The closure property: A set has closure under an operation if the result of combining ANY TWO elements under that...

I think this be Analysis, I Need some kind of convergence theorem for integrals taken over sequences of sets, know one? Example, a double integral taken over sets such that x^(2n)+y^(2n)<=1 with some integrand. I'd be interested in when the limit of the integral over the sequence of sets is...

[b[1. Homework Statement [/b] ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| \ge 8*6^x(8^{x-1}+6^x)## The sets containing the real solutions for some numbers ##a, b, c, d,## such that ##-\infty < a < b < c < d < +\infty## is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)##. Prove it by...

Homework Statement i'm feeling that i didn't quite catch the whole concept of inertial forces very well , and I'm looking for an additional source for mechanics problems . so far i have been learning and solving problems for "an introduction to mechanics" by danniel kleppner , which is btw is...

$P(A \cap B) = P(A) \cap P(B)$ How can we prove this to be true?

I don't see how this is the case. Let ao and bo be members of [A,B] with ao<bo. Let {ai} be a strictly decreasing sequence, with each ai>A and {bi} be a strictly increasing sequencing with each bi<B. Let the limits of the two sequences be A and B, respectively. Then define Ii = [ai,bi]. It seems...

Let $d$ be a metric on $X$. Fix ${x}_{0}\in X$. Let ${d}_{\lambda}\left(x,y\right)=\frac{1}\lambda{}\left| x-y \right|$ and The two sets ${X}_{w}={X}_{w}\left({x}_{0}\right)=\left\{x\in X:{d}_{\lambda}\left(x,{x}_{0}\right)\to0\left( as \lambda\to\infty\right) \right \}$ and...

Homework Statement If there are two sets of matrices ##S = \begin{Bmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix} | a, b, c, d \in \mathbb{C} \end{Bmatrix} ## and ##M = \begin{Bmatrix} \begin{bmatrix} a & b \\ -\overline{b} & \overline{a} \end{bmatrix} | a, b \in \mathbb{C} \wedge |a|...

Homework Statement Let ##A,B \in {\cal P}(E)##. Solve in ##{\cal P}(E)## the following equations: ##X\cup A = B## ##X\cap A = B## ##X - A = B## Homework EquationsThe Attempt at a Solution We have ##A\cup B = (A\cup X)\cup A = A\cup X = B##. So ##A\subset B## and the solution cannot be less...

If S={a,b,c}, what does it mean that S=T when T={a,a,a,a,a,a,b,c}? The mapping that confirms the definition of equality assumes that the duplicate symbols in a set are representative of the same entity or idea. If S={1/2} and T={ .5 , 2/4 , ,25/,5 , 4/8 } are these sets equal? At what point...

Homework Statement given set C = {(x,y)|x,y are integers, x^2 + |y| <= 2} Suppose they are uniformly distributed and we pick a point completely at random, thus p(x,y)= 1/11 Homework Equations Listing it all out, R(X) = {-1,-2,0,1,2} = R(y) The Attempt at a Solution My problem is that when I...

Homework Statement Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ. Is ε closed under countable unions? Homework Equations Potentially De Morgan's laws? The Attempt at a Solution Hi everyone, Thanks in advance for looking at my...

Homework Statement Give an element-wise proof for the following: If A⊆B and B⊆C', then A ∩ C = ∅ Homework Equations A is a subset of B (written A ⊆ B) if every element in the set A is also an element in the set B. Formally, this means that fore every x, if x ∈ A, then x ∈ B. A ∩ B = { x ∈ U ...

Homework Statement Prove that if A and B are sets, then (A - B) U B = A U B I think I might be missing a few steps here. Homework EquationsThe Attempt at a Solution (A - B) U B = 1. (A ^ ~B) U B = 2. (A ^ ~B) U (A ^ B) = 3. A U B

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help...

Homework Statement Consider the sets ##A = \left\{(x_1,x_2) \in\mathbb{R}^2: x_1+x_2 \leq 1\right\}## which is a straight line going through ##(0,1)## and ##(1,0)## and ##B = \left\{(x_1,x_2) \in\mathbb{R}^2: (x_1-3)^2+(x_2-3)^2 \leq 1 \right\}## which is a circle of radius ##1## centred at...

Homework Statement Is the set ##W## a subspace of ##\mathbb{R}^{3}##? ##W=\left \{ \begin{bmatrix} x\\ y\\ z \end{bmatrix}:x\leq y\leq z \right \}## Homework EquationsThe Attempt at a Solution I believe the set is indeed a subspace of ##\mathbb{R}^{3}##, since it looks like it will satisfy...

Homework Statement Let {Ei: 1≤i≤n} be a finite family of closed sets. Then ∪i=1n Ei is closed. Homework Equations Noting that (Ei)c is open The Attempt at a Solution Honestly, I have no idea where to start. I tried to demonstrate that Eai≥Ei if a is a constant greater than zero. Then...

Hello! (Wave) The following definition is given: A set $U \subset \mathbb{R}^n$ is called open if for each $x \in U$ there is $B_d(x, \epsilon) := \{ y \in \mathbb{R}^n: d(x,y)< \epsilon\}$ -> open ball with center $x$ and radius $\epsilon$. Could you explain me why the following set is open...

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need someone to help...

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need help to get...

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need help to get...

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ... At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ... I need help to get...

The formal way to define many mathematical objects is careful not to assert the uniqueness of the object as part of the definition. For example, formally, we might define what it means for a number to have "an" additive inverse and then we prove additive inverses are unique as a theorem...

Homework Statement Let A1,A 2,...be an infinite sequence of events such that A1⊂A2⊂... Prove that Pr(∪Ai)=limn→infPr(An) ∪ is also an iterator that starts from i=1 to infinity. How can you put those iterators?[/B] Homework Equations I decided to use induction The...

Homework Statement In a certain city, three newspapers A, B, and C are published. Suppose that 60 percent of the families in the city subscribe to newspaper A, 40 percent of the families subscribe to newspaper B, and 30 percent subscribe to newspaper C.Suppose also that 20 percent of the...

Homework Statement Let ##A, B, C## be sets with ##A \subseteq B##. Show ##(A-B)\cup C=(A\cup C)-(B\cup C)## Homework Equations None. The Attempt at a Solution So, generally, one shows two sets to be equal by showing that each is a proper subset of the other. I started with the LHS. Thus...

Prove that if X\Y and Y\X are equipotent sets then |X| = |Y|. The problem is that I've no clue where to start... (Futile) attempt: There is bijection $f: X\backslash Y \to Y\backslash X$. For every $r_1 \in X\backslash Y$ there exists $r_2$ s.t. $r_2 \in Y\backslash X$. That's $r_1 \in X$ and...

Homework Statement Let A and B be two events such that P(A) = 0.4, P(B) = 0.7, P(A∪B) = 0.9 Find P((A^c) - B) 2. Homework Equations I can't think of any relevant equations except maybe the Inclusion Exclusion property. P(A∪B) = P(A) + P(B) - P(A∩B) This leads us to another thing P(A∩B^c)...

I was some youtube videos and i got sucked into this channel called "numberphile". They were talking about infinite sets. In particular the set that is the sum of all natural numbers. Through some creative algebra they demonstrate the proof. Somehow the set that is equal to the sum of all...

I read somewhere that mathematical functions can be implemented as sets by making a set of ordered tuples <a,b> where a is a member of A and b is a member of B. That should create a function that goes from the domain A to the range B. But set theory has functions too, could they be sets too...

I want to purchase the Logitech z623 and add to it two creative inspire M2600 speakers (only the speakers, not the sub-woofer). The reason for doing this is because the Logitech system sounds really nice, it has an amazing bass, but mids and highs are not the best and I wish they was as clear as...

Hello everyone. I am looking for some free software to produce a Julia Set that would allow me to enter the equation. I would prefer it to be downloadable software, but if a web applications all there is that will do. It's preferable that a color scheme can be chosen, or at least used. Also, I...

So my question here is: the divergence theorem literally states that Let \Omega be a compact subset of \mathbb{R}^3 with a piecewise smooth boundary surface S. Let \vec{F}: D \mapsto \mathbb{R}^3 a continously differentiable vector field defined on a neighborhood D of \Omega. Then...