Set Definition and 1000 Threads

I am studying ode now, and my text has that If y'=f(y/x) Then, setting y/x=u ; y=ux is a way to solve it. I understand the idea, turn orignal form to separable form. But I can't get the differentiation, Book says y'=u'x+u by product rule which I already know. Here my question is why u=y/x that...

Ok, so I read that a tuple is an ordered list of n elements, where n is a non-negative integer. Sometimes I get confused with tuples and partially ordered sets. So if I organized a list of all the airplanes in a particular airport, would that be a tuple, or would that be a set? I'm leaning...

Hello everyone, Which number set does log(0) belong to? Or does it belong to any number sets?

Let $S = \left\{x_1, \ldots, x_k\right\}$ be independent set consisting of $$k$$ elements in a linear space $$V$$ and let $l(S)$ be the subspace spanned by $S$. Then every set of $$k+1$$ elements in $$l(S)$$ is dependent. I've gotten nowhere with this - but I'm posting my attempt nonetheless...

Homework Statement ##|4^{3x}-2^{4x+2}*3^{x+1}+20*12^x*3^x| > 8*6^x(8^{x-1}+6^x)## For some numbers ##a, b, c, d## such that ##-\infty < a <b < c <d < +\infty ## the real solution set to the given inequality is of the form ##(-\infty, a] \cup [b, c] \cup [d, +\infty)## Prove it by arriving at...

So I have to prove "If (AxB)∩(BxA) ≠ ∅, then (A∩B) ≠ ∅." I wanted to prove by changing it's form. P = (AxB)∩(BxA) ≠ ∅ and Q = (A∩B) ≠ ∅ . The conditional statement is P implies Q and the new statement is not P or Q . P → Q = ¬ P∨Q They are equivalent. But how do I negate P? Would it be...

What is the smallest value in the image set of the function: f(x)=3x^2+10x+219

Homework Statement Number 1 Homework Equations I know I should use a Venn diagram. The Attempt at a Solution The statement says that x intersects y so therefore the statement equals x minus y. This Is my attempt at the solution, if you subtract y from the venn diagram you get this partially...

Homework Statement Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π). Homework EquationsThe Attempt at a Solution Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π) (ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...

Write a while loop that prints 1 to userNum, using the variable i. Follow each number (even the last one) by a space. Assume userNum is positive. Ex: userNum = 4 prints: 1 2 3 4 #include <iostream> using namespace std; int main() { int userNum = 0; int i = 0; userNum = 4; //...

Call a nonempty (finite or infinite) set $A\subseteq\Bbb R$ complete if for all $a,b\in\Bbb R$ such that $a+b\in A$ it is also the case that $ab\in A$. Find all complete sets.

Homework Statement "Determine the set of points at which the function is continuous" ƒ(x,y,z) = √y/(√(x2-y2-z2)) Homework Equations ƒ(x,y,z) = √y/(√(x2-y2-z2)) The Attempt at a Solution If y is zero this function would be discontinuous or if the denominator became zero. As much detail as you...

Hello! I'm currently teaching an advanced course in mathematics at high school. The first half treats discrete mathematics, e.g. combinatorics, set theory for finite sets, and some parts of number theory. Next year I would like to change some of the subjects in the course. My question is: Are...

Has anyone else spotted an unusual set of three different integers A, B, & C such that A^n + B^n - C^n = A + B - C > 0 (n > 1 and A x B x C > 0) I leave the reader to see if they can find this set, or to ask me what they are.

Homework Statement Let U is the set of all commuting matrices with matrix A= \begin{bmatrix} 2 & 0 & 1 \\ 0 & 1 & 1 \\ 3 & 0 & 4 \\ \end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...

Hi, While solving a system of linear equations, there are three possible cases - unique / infinite / no solutions - to the system. One geometric interpretation is when one looks at a set of planes intersecting at one / many / no points respectively, for each of the above cases. While going...

Suppose you have a set of vectors v1 v2 v3, etc. However large they are, suppose they span some area, which I think is typically represented by Span {v1, v2, v3} But I mean, if you're given these vectors, is there anything wrong with rearranging them? Because there's a theorem- that "an...

Homework Statement Sorry this may be a better picture. I missed the top of the page but the question says 1) In the pole vault event, an athlete runs as fast as possible towards the bar, holding a flexible fiberglass pole. He stick s the end of the pole into a slot in the ground, swings up on...

Hello! (Wave)We have that $S_{||\cdot||_2}:= \{ x \in \mathbb{R}^n: ||x||_2=1\}$. How can we justify that the above set is bounded? Do we just say that if $x \in S_{||\cdot||_2}$ then $||x||_2=1 \leq 1$ and so the set is bounded. How could we justify it more formally?

Mod note: Moved from technical math section, so there's no template. Sorry if I'm formatting this question wrong, new user. F(x) is an antiderivative of https://upload.wikimedia.org/math/9/1/5/915ca58b070b0328cd069524c2d487f2.pngx3+x+1...

Homework Statement For each structure, draw a directed graph representing the membership relation. Then determine which of the following axioms is satisfied by the structure: Extensionality, Foundation, Pairing, Union U= {a,b} a in b , and b in a The Attempt at a Solution The directed...

Homework Statement Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\} Homework Equations - Span -Subset The Attempt at a Solution...

Homework Statement I'm reviewing my powerpoints from class and see the formula A ∩ B ⊆ A. Is this a correct formula? I interpret this as all elements of set A intersected with set B is a subset of set A. I don't think this is a true statement, is it? Sorry it's been a while since I have...

Homework Statement Which pairing is not a set of conjugates? A. OH1– and H2O B. HC2H3O2 and C2H3O21– C. HCl and Cl1– D. NH3 and NH41+ E. H2SO4 and SO42– Homework Equations - The Attempt at a Solution I couldn't choose because conjugates, from what I learnt, is basically an acid or a base and...

Among the following programs, which are the best for theoretical particle physics? Stony Brook, Penn State, Iowa State, Urbana Champaign, Arizona, Arizona State, Georgia Tech, Florida, Tennesse KnoxVille Are there some other programs of lower ranking in the league tables, but has a strong high...

[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 So I was doing this problem in Munkres's Topology book: Determine whether the statement is true or false, If a double implication fails, determine whether one or the other of the possible implications holds: A ⊂ B or A ⊂ C ⇔ A ⊂ ( B ∪ C ) Homework Equations - The Attempt...

Homework Statement Definition of spanning set: Let be vectors in the vector space . The set of all linear combination of the vectors is a subspace ( say ) of . The subspace is called the space spanned by the vectors The set is called a spanning set of . Definition of linear independence...

Hello! (Wave) We consider the set $K=\{ x \in \mathbb{R}^n: \sum_{j=1}^n |x_j|^2 \leq 1 \}$ and let $ \in \mathbb{R}^n$. Check if the euclidean distance of $z$ from $K$ is attained. I want to show that $K$ is closed and convex. Then by a theorem, the distance will be attained by a unique...

Homework Statement ##\frac{x+\sqrt3}{\sqrt{x}+\sqrt{x+\sqrt3}} + \frac{x-\sqrt3}{\sqrt{x}-\sqrt{x-\sqrt3}} = \sqrt{x}## All real solutions to this equation are found in the set: ##a) [\sqrt3, 2\sqrt3), b) (2\sqrt3, 3\sqrt3), c) (3\sqrt3, 6), d) [6, 8)## Homework Equations 3. The Attempt at a...

Homework Statement If A is a closed set that contains every rational number in the closed interval [0,1], show that [0,1] is a subset of A. Homework EquationsThe Attempt at a Solution I'm confused because for the set A = all rationals in [0,1], every point is a boundary point so the set is...

Homework Statement Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\} Homework Equations -Vector space span -Linear independence...

Dear Group, I just set up a Microsoft Virtual Wifi Miniport Adapter using administration command prompt, its function as wifi hot spot, so the other devices could use this hot spot for the internet. ( ) But whenever I shutdown the computer and turn it on, I can not find it, I have to...

In electromagnetism we have these two Lorentz scalars: ##P=B^2-E^2## ##Q=E\cdot B## WP https://en.wikipedia.org/wiki/Classification_of_electromagnetic_fields claims that these are a complete set of invariants, because "every other invariant can be expressed in terms of these two." How does...

Hey all, student100's brother here, he got me to create an account to ask my question here... I'm taking an introduction to linear algebra class and we had a test problem to prove a set of vectors is linear independent: ##V = v_1, \dots, v_n \in \Bbb R^n## such that each element of the set is...

Homework Statement ##\frac{log_{2^{x^2+2x+1}-1}(log_{2x^2 + 2x + 3}(x^2 - 2x)}{log_{2^{x^2+2x+1}-1}(x^2 + 6x + 10)} \geqslant 0 ## The set of all real solutions to this inequality is of the form: ##(a) ## ##(a,b) \cup (b,c) ##, ##(b) ## ##(-\infty,a) \cup (c,\infty) ##, ##(c) ## ##(a,b) ## for...

The following is from an introduction to groups. It is not clear to me why the authors bothered to introduce the subset ##\mathfrak{Q}\subseteq \mathfrak{R}## and a subset ##\mathfrak{K}\subseteq \mathfrak{S}^{\mathfrak{R}}## into the discussion. (3) seems to follow trivially from the...

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Replacement which reads as shown...

I am reading Micheal Searcoid's book: Elements of Abstract Nalysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Subset Principle which reads as shown below...

After I installed a fresh windows server 2012R, I could log in with administrator account, e.g MYWIN\Administrator (MYWIN is my computer name). Then I created a DNS/forest e.g ABC.com , I then could log in with ABC\Administrator. For some reason I now sign out and would want to log in with...

Cardinality of the set of binary-expressed real numbers This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis. 1. Counting...

Homework Statement Find a recursive relation on the number of partitions ##P_n## for a set ##S_n## of cardinal ##n##. ##P_0 = 1## is given. Homework EquationsThe Attempt at a Solution A partition of ##S_{n+1}## is given by the choice of a non-empty ##k##-block ##A_k## of ##S_{n+1}## and a...

Homework Statement Suppose E1 and E2 are a pair of compact sets in Rd with E1 ⊆ E2, and let a = m(E1) and b=m(E2). Prove that for any c with a<c<b, there is a compact set E withE1 ⊆E⊆E2 and m(E) = c. Homework Equations m(E) is ofcourese referring to the outer measure of E The Attempt at a...

Does the set of random integers with random lengths (the number of digits), which hypothetically would generate random numbers with random lengths for eternity, produce all possible integers? It seems to me that this is a natural conclusion but I've never seen a proof of this. A more incredible...

We can easily find out below rules in set theory: 1. Let consider set “A” as follows: A = {a1, a2, a3, a4… an} and also power set of A is set C: C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}} Rule 1: To find the number of subsets with precise members number, we can use Binomial...

Homework Statement \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} Is this set a subspace of ℝ3 Homework Equations The set must be closed under addition. The set must be closed under multiplication. The set must contain the zero vector. The Attempt at a...

Homework Statement use triple integral to find volume of the solid in the first octant that is bounded above by x+2y+3z=6 and laterally by the clyinder x^2+y^2=4 Homework EquationsThe Attempt at a Solution From the given plane I got: Z=2-\frac{1} {3}x-\frac{2} {3}y from the given Cylinder i...

Homework Statement use spherical coordinates to find volume inside a sphere of radius 4 and outside sphere of radius 2, and inside/above the cone z=3√(x^2+y^2). Homework Equations [/B] x=rcosθ =ρsinφcosθ , y=rsinθ =ρsinφsinθ z=ρcosφ r= ρsinφ The Attempt at a Solution replacing z^2 in the...

Hi All, A simple question but one for which I cannot seem to get the intuition. 1. Homework Statement Find the interior point of {1/n : n ∈ ℕ}. Homework Equations N/A The Attempt at a Solution Let S = {1/n : n ∈ ℕ}, where S ⊆ℝ x is an interior point if ∃N(x ; ε), N(x ; ε) ⊆ S. My...

I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...