Set Definition and 1000 Threads

In the Zermelo-Fraenkel axioms of axiomatic set theory we find: Axiom. Given any set x, there is a set such that, given any set z, this set z is a member of if and only if every element of z is also an element of x. Why is this needed as an axiom? why isn't it merely a definition? Under...

Hi, I'm reading a paper about combustion and I can't understand the composition of the inlet mixture. Here's what the paper says: "A known flow rate of the isobutene (Matheson, TriGas, 99.5%) in nitrogen (General Air, 99.998%) was introduced into a 6mm ID quartz reactor that was heated in an...

I don’t see how the definition of |an> transmorphs into the statement involving the kroneck delta functions.

Dear Physics Forums members, I have a research problem that involves electrostatics. My education is as a chemist, and thus I struggle to accurately represent my problem, so I thought that you guys could help me (and would be interested in the exercise). Here is an image to summarize my...

Homework Statement Let ##K\neq\emptyset## be a compact set in ##\Bbb{R}## and let ##c\in\Bbb{R}##. Then ##\exists a\in K## such that ##\vert c-a\vert=\inf\{\vert c-x\vert : x\in K\}##. 2. Relevant results Any set ##K## is compact in ##\Bbb{R}## if and only if every sequence in ##K## has a...

Problem: Let ##G=S_n##, fix ##i \in \{1,2, \dots, n \}## and let ##G_i = \{ \sigma \in G ~|~ \sigma (i) = i \}##. Use group actions to prove that ##G_i## is a subgroup of G. Find ##|G_i|##. So here is what I did. Let ##A = \{1,2, \dots, n \}##. I claim that ##G## acts on ##A## by the group...

$$\tiny{la1.1.22}$$ $\textsf{Construct 3 different augmented matrices for linear systems whose solution set is}$ $$x_1=3,\quad x_2=-2,\quad x_3=-1$$Well we could start just by$$3x-2y+y=-1$$but then we need a $3\times4$ matrix

I'm thinking the term is congruent. For example, let's say there are 2 parametric lines: s0( t ) = < 2 , 1 > t + < 0 , - 3 > s1( t ) = < -18 , -9 > t + < -6 , -6 > The loci for both are the same line, just that one proceeds in the opposite as the other (and at a high rate) per change...

Hi, I would like to implement voltage regulator based on microcontroller + triac. I am able to calculate source RMS voltage by sampling signal using ADC, but don't know how to calculate triac angle to set desired RMS voltage. For example i have 250v source RMS voltage and i need to calculate...

Find the elements in each set given that: U={1,2,3,4,5,6,7} A={1,3,5,7} B={2,4,6} C={1,2,5,6} D={2,3,4} A) C intersect D B) C-D C) (A union B) union C D) (C union A)' -D

Homework Statement Proposition. Every subset of a finite set is finite. 2. Relevant definitions Definition. Two sets ##X## and ##Y## have the same cardinality iff there is a bijection between ##X## and ##Y##. A set ##X## is finite iff there is a bijection between ##X## and ##\{1, ... , n\}##...

Hi everybody, I have a problem with this question: Choose from the following set of vectors in R4 set of the maximum number of linearly independent vectors: 1.(1;2;3;4) 2.(2;4;6;8) 3.(0;1;-1;1) 4.(1;1;1;0) 5.(0;3;0;0) So, I can find a rank of a matrix, it's 4, but I can't understand how it...

I am trying to prove that any subset of a countable set is either finite or countable. I know that a set S is countable if there exists a bijection that takes S to \Bbb{N}. My first thought was to consider the subset V of S. If V is finite we are done, since we can always construct a finite...

Let f:\mathbb{R}^m\rightarrow\mathbb{R}^m. Define the zero set by \mathcal{Z}\triangleq\{x\in\mathbb{R}^m | f(x)=\mathbf{0}\} and an \epsilon-approximation of this set by \mathcal{Z}_\epsilon\triangleq\{x\in\mathbb{R}^m|~||f(x)||\leq\epsilon\} for some \epsilon>0. Clearly \mathcal{Z}\subseteq...

I previously posted the below as a footnote on one my posts on the Mechanical Engineering Forum and it was suggested I start this thread on the subject here. "PS On a general note about battery replacements. On my field GPS I use rechargeable batteries and always test them before recharging...

I would like to solve this system, which is a sets of non linear quadratic equations, the system needed to be solved can be expressed in general as follow: ϒϒ'C – ϒα = B Where ϒ=(ϒ1,ϒ2,...ϒn)’ is a column vector and ϒ’ its transpose C=(c1,c2,…,cn)’ and B=(b1,b2,…bn)’ are a columns vector And...

Homework Statement 2. Let ##T: \mathbb{P}^1 \rightarrow \mathbb{R} \text{ be given by } T(p(x)) = \int_0^1 p(x)~dx## Describe ##Ker(T)## using set notation. Homework Equations ##p(x) \in \mathbb{P}^1~|~ p(x) = a_0 + a_1x## The Attempt at a Solution ##T: \mathbb{P}^1 \rightarrow \mathbb{R}##...

So apparently the proof involves a trick that converts the problem of a general power set ##\mathscr{P}(M)## of some set ##M## which has of course the property of not having pairwise disjoint set-elements to a problem that involves disjoint set-elements. I do not understand why this trick is...

Homework Statement Write out the following set by listing its elements between braces: {X : X ⊆ {3,2,a} and |X| = 4} Homework EquationsThe Attempt at a Solution My book tells me that it is ∅, but couldn't X have repeated elements? so X = {3,2,a,3}, {3,2,a,2} etc for example??

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.1 Generating and Cogenerating Classes ... ... I need some help in order to make a meaningful start on Problem 2, Problem Set 4.1 ... Problem 2, Problem Set 4.1 reads as follows:( *** NOTE ...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need help to make a meaningful start on Problem 14 of Problem Set 2.1 ... Problem 14 of Problem Set 2.1 reads as follows:I am somewhat overwhelmed by...

Homework Statement {(x,y) ∈ ℝ^2 : (y-x)(y+x) = 0 } Homework EquationsThe Attempt at a Solution [/B] Is the reason the answer is the union of y = -x and y = x because if you divide (y-x)(y+x) = 0 by y-x you get y = -x and if you divide (y-x)(y+x) = 0 by y+x you get y = x? and if you...

Greetings, I have an experiment that pressurizes a small volume at variable rates. 1, 2, and 3 psi/sec. I am trying to think of a way to reproducibly drop the pressure of the container by a known amount while it is being pressurized. Are there valves for this type of experiment? What are...

Homework Statement Let ##J = \emptyset## and ##A## be any set. Then ##J \subseteq A##. Homework EquationsThe Attempt at a Solution Proof: Suppose that it is false that ##J \subseteq A##. Then ##\exists x (x \in J ~~\text{and}~~ x \not \in A)## is true. But this is a contradiction, since...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ... I need some help in order to make a meaningful start on Problem 1, Problem Set 4.1 ... Problem 1, Problem Set 4.1 reads as follows: Can someone...

Prove in any axiomatic set theory that:2ε{a,2,b} , where a,b are letters

Homework Statement I don't want to clog up the forums with a few "small" problems so I am lumping them together here. 2. Let ##T:P^1 → R \text { be given by } T(p(x)) = \int^b_a p(x)dx##. Describe Ker(T) using set notation. 3. Let ##H = \left\{f ∈ C[a, b] | f'(x) ≥ 0 ~\text for...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.2 Free Modules ... ... I need someone to check my solution to the first part of Problem 1(a) of Problem Set 2.2 ... Problem 1(a) of Problem Set 2.2 reads as follows: My solution/proof of the...

Homework Statement I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.2 Free Modules ... ... I need someone to check my solution to the first part of Problem 1(a) of Problem Set 2.2 ... Problem 1(a) of Problem Set 2.2 reads as follows: Can...

A number of classicists were asked to identify their favorite epic poets. The results are summarized below. 62 appreciate Homer, 40 appreciate Virgil, 5 appreciate Gomer, 2 appreciate Homer and Virgil and Gomer, 3 appreciate Virgil and Gomer, 20 appreciate Homer and Virgil, 42 appreciate Homer...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need someone to check my solution to Problem 3(a) of Problem Set 2.1 ... Problem 3(a) of Problem Set 2.1 reads as...

Homework Statement I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need someone to check my solution to Problem 3(a) of Problem Set 2.1 ... Problem 3(a) of Problem Set 2.1 reads as follows: Homework...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need someone to check my solution to Problem 2(c) of Problem Set 2.1 ... Problem 2(c) of Problem Set 2.1 reads as follows:My attempt at a solution...

Homework Statement I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need someone to check my solution to Problem 2(b) of Problem Set 2.1 ... Problem 2(b) of Problem Set 2.1 reads as follows: Homework...

I request your help in order to know, how can i configure this problem as a continuous markov chain, need to define the main variable, the states, transition rates, and the matrix. I thought that it could be relationed with the independent status of the machines, because if the machine 1 is...

Two stationary masses will attract each other. That is, they start moving towards each other. For example, a stationary apple will fall to earth. I can see bent space affecting the trajectory of a moving object. ...but how does bent space explain that stationary objects start moving?

Homework Statement For the following notation, D \subseteq \{ z | \exists x,y \in S (z = x - y ) \} I'm wondering if S can only have one element, or does the notation imply that | S | > 1 Homework EquationsThe Attempt at a Solution For example, if I have S = \{ 3 \} is z = 3 - 3? Or...

Hello. I self-studied and have a good grasp on QM, statistical mechanics and Group theory. Next step is QFT. There are several sets of video lectures on Youtube about this subject and I am asking for a recommendation (I would like a set of videos which involves the unitary IR of the Poincare...

Hi, I have a website and I would like to give viewers an option for downloading a (Mathematica) file. Actually I have a blog in blogger.com which has a menu item to access an index.html file on another hosted website. I would like to store the file on the hosted website and have a button or...

Homework Statement So I have the set {...(1/8),(1/4),(1/2),(1),(2),(4),(8)...} I am suppose to put it in set builder notation.. Homework EquationsThe Attempt at a Solution my answer was { x = {\frac {1}{2^n} : n \in ℤ } } but my books was { x = {{2^n} : n \in ℤ } } I understand both...

Given that A = {1, 2, 3 ……..20} and R is a ternary relation on A defined by equation x2 + 4y = z .Find the set of ordered triples of R.

I want a way to map natural numbers to possible values of set A. It would also be helpful if you could tell me how many possible values are for set A(it depends of k). All that is known about set A is: Non of set A elements is subset to another element of A, but all elements of set A are proper...

Could someone please check if these are right?

Hypothetical question: Let's say there's a granite cube with a volume of 60 cubic centimeters. A cannon ball is shot at it, reducing it to rubble, and the kinetic energy of the ball was 10,000 joules. We use the same ball with the same kinetic energy output on a stronger unknown substance with...

Hi PF, I have a setup with two plates that are connected to two different voltages and separated by vacuum. In my case, one plate is set at -10kV, and the other plate is set at -7.5kV. I know the area of these plates, the separation of the plates (which is much smaller than the plate area), and...

Hello, could you please help me regarding this question about a certain map (application). Homework Statement Demonstrate if A1 ⊂ A2 →ƒ(A1) ⊂ ƒ(A2) 2. Homework Equations ƒ:A→B is a map A1, A2⊂ A The Attempt at a Solution first assumed that (A1∪A2)⊆A ⇒ (ƒ(A1) ∪ ƒ(A2))⊆ ƒ(A) then if A1 ⊂ A2 ∴...

Homework Statement Show that {(1, 2, 3), (3, 4, 5), (4, 5, 6)} does not span R3. Show that it spans the subspace of R3 consisting of all vectors lying in the plane with the equation x - 2y + z = 0. Homework EquationsThe Attempt at a Solution I made a matrix of: A = [ 1 3 4 ; 2 4 5; 3 5 6] and...

Hello all, I am an undergraduate student who is studying real analysis from Rudin's POMA and I am trying to prove that these two definitions that I have for dense sets are equivalent: 1) Given a metric space X and E ⊂ X ; E is dense in X iff every point of X is a limit point of E or E = X or...

Homework Statement Two point charges are placed on the x axis.(Figure 1)The first charge, q1 = 8.00 nC , is placed a distance 16.0 m from the origin along the positive x axis; the second charge, q2 = 6.00 nC , is placed a distance 9.00 m from the origin along the negative x axis. Find the...

Recently I posted the following text on the Analysis forum ..." ... ... In order to demonstrate that K is closed, we prove that \mathbb{R}^n\K is open. Indeed, choose y \notin K and define O_j = \{ x \in \mathbb{R}^n \ \mid \ \mid \mid x - y \mid \mid \gt \frac{1}{j} \} for j \in \mathbb{N}...