Set Definition and 1000 Threads

Can someone explain me,why Wheatstone bridge is most sensitive when all four resistances say A,B,C and D are equal?as far as i know condition for Wheatstone Bridge is A/B=C/D.

So I'm reading up on some set theory, and I came to the axiom of pairing. The book uses that axiom to prove/define a set which contains the elements of two sets and only the elements of those two sets. ##~~B## is the set which contains the elements and only the elements of sets ##a## and...

Let $A$ be the set of all positive integers $a$ such that $2^{2008}+2^a+1$ is a square. Find the smallest number of $A$ and prove that it is not the only member of $A$.

Homework Statement Prove or disprove the following (i) ##\forall a\in\mathbb{R}[(\forall \epsilon>0,a<\epsilon)\Leftrightarrow a\leq 0]## 2. The attempt at a solution Can't we disprove the above statement by telling ##a\leq 0 \nRightarrow (\forall \epsilon>0,a<\epsilon)## through a counter...

Please help me to define correctly, in the language of mathematics, the configuration of sets shown on the picture. Homework Statement I'd like to define the following rules: U is a set with infinite members. L is a list or set of properties. Every property (Ls1, Ls2 ... ) have a value (...

I am considering the following question and I want you to agree (but perhaps you don’t):Rutherford wrote a letter to Bohr, as an answer to a previous letter from Bohr containing one of the first of Bohr’s descriptions of the atomic model, saying that he understood the atom model Bohr advocated...

I'm reading the proof that a cauchy sequence is convergent. Let an be a cauchy sequence and let ε=1. Then ∃N∈ℕ such that for all m, n≥N we have an-am<1. Hence, for all n≥N we have an-aN<1 which implies an<aN+1. Therefore, the set {n∈ℕ: an≤aN+1} is infinite and thus {x∈ℝ : {n∈ℕ: an≤x} is...

Whats the difference between a level set and a level curve, with regards to multi-variable calculus?

I'm coming from a physics background, but find pure mathematics extremely interesting, so have decided to try and gain a more fundamental understanding of the subject. I've recently been reading up on relations and how one can define them as sets of ordered pairs. I am particularly interested in...

Homework Statement My teacher has notes online that say: A Simple Construction Technique for WellFounded Orders Any function ƒ : S→N defines a wellfounded order on S by x < y iff ƒ(x) < ƒ(y). Example: Lists are wellfounded by length. Binary trees are wellfounded by depth, by number of nodes...

Homework Statement Find the largest set D on which f(z) is analytic and find its derivative. (If a branch is not specified, use the principal branch.) f(z) = Log(iz+1) / (z^2+2z+5) Homework EquationsThe Attempt at a Solution Not sure how to even attempt this solutions but I wrote down that...

Homework Statement Homework EquationsThe Attempt at a Solution I have managed to solve it for the finite case, where the masure is less than infinity. But how do I solve it if the ,measure if the measure of E is infinite?

Homework Statement Prove that set of all onto mappings of A->A is closed under composition of mappings: Homework Equations Definition of onto and closure on sets. The Attempt at a Solution Say, ##f## and ##g## are onto mappings from A to A. Now, say I have a set S(A) = {all onto mappings of A...

The class is called Math for EE and CE. The professor teaches from his own notes and doesn't give many examples. Any help checking my work would be appreciated and/or if you could point me in the direction of more examples like these. I've looked trough Set Theory and discrete math books but...

Homework Statement Assume that y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on an open interval I on which p,q are continuous. Assume also that y1 and y2 have a common point of inflection t0 in I. Prove that y1,y2 cannot be a fundamental set of solutions unless p(t0) = q(t0) = 0.The...

I am working on this problem on measure theory like this: Suppose ##X## is the set of real numbers, ##\mathcal B## is the Borel ##\sigma##-algebra, and ##m## and ##n## are two measures on ##(X, \mathcal B)## such that ##m((a, b))=n((a, b))< \infty## whenever ##−\infty<a<b<\infty##. Prove that...

Homework Statement Show that the set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 is closed but not compact. Homework Equations set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 The Attempt at a Solution I set x = 0 and then y = 0 giving me [0,±√3] and [±√3,0] which means it is closed However, for it to...

My gut instinct is that putting all the resistors in series will give max resistance, but I am not sure how to give a more rigorous either mathematical or just in words reasoning why. Or maybe I'm wrong! But it seems like the fraction introduced from parallel won't help In any case, assume all...

I have the hardback 5 volume set of Spivak's A Comprehensive Introduction to Differential Geometry that is in pretty good shape. Is there any value to that set? I tried looking it up, but I don't really see many people selling whole sets, so I can't tell... Thanks.

Homework Statement Find a set of basic solutions and express the general solution as a linear combination of these basic solutions a + 2b - c + 2d + e = 0 a + 2b + 2c + e = 0 2a + 4b - 2c + 3d + e = 0 Homework Equations 3. The Attempt at a Solution [/B] i reduced it to: 1 2 0 0 -1 0 0 0 1...

Having trouble understanding the concept of transitivity. By definition: If (a,b)\in R\wedge (b,c)\in R \Rightarrow (a,c)\in R - Great. Consider the set \{a,b\}. What makes the relation \{(a,a)\} or \{(a,a),(a,b)\} transitive? How do I translate this in terms of the definition? What makes an...

Let $f:\mathbf R^n\to\mathbf R^m$ be a smooth function of constant rank $r$. Let $\mathbf a\in \mathbf R^n$ be such that $f(\mathbf a)=\mathbf 0$. Then $f^{-1}(\mathbf 0)$ is a manifold of dimension $n-r$ in $\mathbf R^n$. We imitate the proof of Lemma 1 on pg 11 in Topology From A...

A theorem on equivalence relation states that for any set S, the set of equivalence classes of S under an equivalence relation R constitutes a partition of a set. Moreover, given any partition of a set, one can define an equivalence relation on the set. What allows you to "create" a partition...

Homework Statement D1 = {(x,y) : x^2 + y^2 < 3, x+2y = 2} D2={(x,y) : x^2 + y^2 > 2} D3={(x,y) : x + 2y = 2} Homework EquationsThe Attempt at a Solution D1 is neither, D2 is open and D3 is closed, am I right or wrong?

Homework Statement R = all real numbers F(x) = { y in R : sin(y) = x} 1. Is F a mapping from R to R 2. Describe the three sets F(5), F(0), F(1). 3.Can F be represtented as a function from R to R 4. Give two different choices of X and Y (take both X and Y to be subsets of ?) so that F can...

Hello! (Wave) Suppose that $X$ contains a countable set. Let $b \notin X$. Show that $X \sim X \cup \{b\}$. Prove that in general if $B$ is at most countable with $B \cap X=\varnothing$ then $X \sim X \cup B$. Proof:We will show that $X \sim X \cup \{b\}$. There is a $\{ a_n: n \in \omega \}...

Hello! (Smile) Proposition: The set $\{0,1\}^{\omega}$ of the finite sequences with values at $\{0,1\}$ is not countable. Proof: $$\{ 0,1 \}^{\omega}=\{ (x_n)_{n \in \omega}: \forall n \in \omega \ x_n \in \{0,1\} \}$$ From the following theorem: No set is equinumerous with its power set...

Hello! (Smirk) Proposition The set $\mathbb{Z}$ of integers is countable. Proof $\mathbb{Z}$ is an infinite set since $\{ +n: n \in \omega \} \subset \mathbb{Z}$. $$+n= [\langle n, 0 \rangle]=\{ \langle k,l \rangle: k+n=l\}$$ We define the function $f: \omega^2 \to \mathbb{Z}$ with...

Hello! (Wave) I want to show that if $A$ is a finite set of finite sets then the set $\bigcup A$ is finite. The set $A$ is finite. That means that there is a natural number $n \in \omega$ such that $A \sim n$, i.e. there is a bijective function $f$ such that $f: A...

Hello! (Wave) I am looking at the proof of the following proposition: Let $X \neq \varnothing$. $X$ is at most countable iff there is a $f: \omega \overset{\text{surjective}}{\rightarrow} X$. Proof: "$\Rightarrow$" If $X$ is infinite, then obviously there is a $f: \omega...

Homework Statement Show if the set is convex or not! S2 = Homework Equations I know that to show a set is convex you can either use the definition or show that the set can be obtained from known convex sets under operations that preserve convexity. Convex definition: x1*Theta + (1 -...

Homework Statement Problem: Given a regular deck of 52 cards, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}. Describe the event A ∪ B Solution: A ∪ B = {either king or club or both (where "both" means "king of clubs")} Homework Equations...

Homework Statement Describe the set of points determined by the given condition in the complex plane: |z - 1 + i| = 1 Homework Equations |z| = sqrt(x2 + y2) z = x + iy The Attempt at a Solution Tried to put absolute values on every thing by the Triangle inequality |z| - |1| + |i| = |1|...

Let X be a real Banach Space, C be a closed convex subset of X. Define Lc = {f: f - a ∈ X* for some real number a and f(x) ≥ 0 for all x ∈ C} (X* is the dual space of X) Using a version of the Hahn - Banach Theorem to show that C = ∩ {x ∈ X: f(x) ≥ 0} with the index f ∈ Lc under the...

I am doing some self study of groups and can solve problem #3 but not Problem #4. Problem 3. Let A be a finite set, and B a subset of A. Let G be the subset of S_A consisting of all of the permutations f of A such that f(x) is in B for every x in B. Prove that G is a subgroup of S_A. Problem...

Proposition: The set $\omega \times \omega$ is equinumerous with $\omega$, i.e. the set $\omega \times \omega$ is countable. "Intuitive Proof" $$\mathbb{N}^2=\{ (n,m): n,m \in \mathbb{N} \}$$ $$1 \mapsto a_{11}$$ $$2 \mapsto a_{12}$$ $$3 \mapsto a_{31}$$ $$4 \mapsto a_{22}$$ $$5 \mapsto...

I think we have a problem with this rule. This is my understanding of the existing rule. Am I close? The rule for calculating the power distribution on a power split planetary gear system is based on the equal force rule. To calculate the power on the sun and ring of a planet set with input...

Homework Statement A rod of length ##L## and mass ##M## is constrained to move in a vertical plane. The upper end of the rod slides freely along a horizontal wire. Let ##x## be the distance of the upper end of the rod from a fixed point, and let ##\theta## be the angle between the rod and the...

Hello, I am just doing my homework and I believe that there is a fault in the problem set. Consider the set of functions defined by V= f : R → R such that f(x) = a + bx for some a, b ∈ R It is given that V is a vector space under the standard operations of pointwise addition and scalar...

Suppose you have two observables ##\xi## and ##\eta## so that ##[\xi,\eta]=0##, i know that there exists a simultaneous complete set of eigenvectors which make my two observables diagonal. Now the question is, if ##\xi## is a degenerate observable the complete set of eigenvectors still exist?

Hi! (Smile) According to my notes, a set $A$ is called transitive if the elements of its elements are elements of $A$. For example, the set of natural numbers $\omega$ is a transitive set. Also, if $n \in \omega$ then $n$ is a transitive set since $n=\{0,1,2, \dots, n-1 \}$ and if we take a...

Are these three sets equivalent? $$A=\left\{(x,y):x,y\in\Bbb{R},y\ge x^2-1\right\}$$ $$B=\left\{x,y\in\Bbb{R}:y\ge x^2-1\right\}$$ $$C=\left\{(x,y)\in\Bbb{R}^2:y\ge x^2-1\right\}$$ I am thinking that $A$ and $C$ are, but not $B$ as it might be ambigious as to which dimension it is in, i.e it...

Homework Statement The problem is to prove that there is only one empty set. Let A and B be empty sets, A is a subset of B and B is a subset of A (by the definition that the empty set is a subset of every set) So A=B (by definition) By convention, all empty sets are equal. Therefore, there...

Homework Statement Bob makes two sets: one with all the even integers between 1 and 30 inclusive, and another with all the odd integers inclusive. He called the sets Q and R. He multiplied each number from Q with each number in R. Then he added the 225 products together and called the result...

Hello! (Wave) According to my notes, when we consider the order $I_A$ for $A \neq \varnothing$ each element of $A$ is minimal and maximal. If, in addition, $A$ has at least two elements then there isn't neither the greatest nor the least element of $A$. Could you explain it to me? :confused...

Let's see if I have this straight: Observables are represented by Hermitian operators, which can be, for some appropriate base, represented in matrix form with the eigenvalues forming the diagonal. Sounds nice until I consider observables with continuous spectra. How do you get something like...

I'm introducing myself to set theory. My reference doesn't seem to address the fact that 1/1 = 2/2 = 1. If we make a correspondence between natural numbers and rational numbers using sequential fractions, should we just skip equivalent fractions so as to make it a bijection? In other words, does...

Homework Statement Shown in attachment The problem has been modified. All inputs and outputs are 5 gal/min. Pure water enters tank 1. Homework Equations System of equations The Attempt at a Solution Included on attachment.

Hello! (Wave) I want to describe an algorithm with time complexity $O(m)$ that, given a set $M$ with $m$ numbers and a positive integer $p \leq m$, returns the $p$ closest numbers to the median element of the set $M$. How could we do this? (Thinking)

HI all, I have the equation, 6x^2-2>9x for which I'm to find the solution set in interval notation. I've rewritten the inequalty as 6X^2-9x-2=0. I tried to factor, but no go. Then I used the quadratic and got 9+/- rad(129)/-18. The answers I get for x are -1.1309 and .1309. The calculator...