What is Set: Definition and 1000 Discussions

Seth, in Judaism, Christianity, Mandaeism, Sethianism, and Islam, was the third son of Adam and Eve and brother of Cain and Abel, their only other child mentioned by name in the Hebrew Bible. According to Genesis 4:25, Seth was born after Abel's murder by Cain, and Eve believed that God had appointed him as a replacement for Abel.

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  1. L

    If all elements of a set are individually bounded, is the set bounded?

    This is a concise question, so the title pretty much says it all. Also, this is not a HW question, but the idea has subtly popped up in two homework problems that I have done in the past. I cannot justify why the entire set would be bounded, because we know nothing of the nature of the...
  2. A

    Describe the closure of the set with formulas

    Homework Statement -∏<arg(z)<∏ (z≠0) Homework Equations arg(z) is the angle from y=0The Attempt at a Solution Arg(z) spans the entire graph since -pi to pi is the full 360 degrees so I put: -∏<arg(z)<∏ --> 0<arg(z)<2∏+k∏, (k ε Z) --> arg(z) \subset R --> arg(z) = R: all real numbers but I...
  3. J

    Calc 1 question? Can you set these two equations equal to each other?

    Find the value(s) of k such that f(x) is continuous everywhere: x^2-7 if x <= 2 and 4x^3-3kx+2 if x>2 Can you set the two equations equal to each other if only one of them has k in it?
  4. B

    Orthonormality of a complete set of eigenvectors

    hello How to you rigorously express the orthonormality of a complete set of eigenvectors (|q\rangle)_q of the position operator given that these are necessarily generalized eigenvectors (elements of the distribution space of a rigged hilbert space)? The usual unformal condition \langle...
  5. S

    Unique combinations in a set.(+)

    Hey there! I will start of with saying I´m not very good at English when considering mathematical terms, neither an expert in Math. My question goes as this: I have a set of 1000 questions - which will be given in rounds with a set of 10. So every round, you get 10 questions out of...
  6. P

    Set Theory: Is {a} a Subset of {S}?

    Homework Statement I am not sure if set theory is precalc or not but here is my question. Find a pair set such that {a} belongs to the set and {a} is not a subset of S. The Attempt at a Solution So I thought that a set like this would work S = {{a}, b} because {a} belongs to the set...
  7. I

    How can the constraint condition be used to define generalized coordinates?

    Homework Statement Build the lagrangian of a set of N electric dipoles of mass m, length l and charge q. Find the equations of motion. Find the corresponding difference equations. Homework Equations Lagrange function L=T-V Lagrange's equations \frac{d}{dt}\left(\frac{\partial L}{\partial...
  8. S

    How do I find the set of solution to x^2z^3 - y^6 over C?

    I'm at a bit of a loss how to do this. I suspect it's the set \left\{t_1^3, t_1t_2, t_2^2) \mid t \in {\mathbb C}\right\} . Certainly the polynomial x^2z^3 - y^6 vanishes on these points, but I'm not sure how to show the other inclusion. The only thing I can think of gets me close, but not...
  9. B

    Reduction Gears: Solving Math Problem for 34mm Dial

    Okay, so I am having a problem, I'm not the greatest at math, here is the problem: I need a set of reduction gears, my plan is to have a gear attached to a shaft (base diameter of this first gear is 33mm) with 50 teeth (maybe?) To turn a reduction gear that will be on another shaft that will...
  10. C

    Topological indistinquisable points and set theory.

    In set theory a set is defined to be a collection of distinct objects (see http://en.wikipedia.org/wiki/Set_%28mathematics%29), i.e. we must have some way of distinguishing anyone element from a set, from any other element. Now a topological space is defined as a set X together with a...
  11. C

    Definition of a circle in point set topology.

    The circle seems to be of great importance in topology where it forms the basis for many other surfaces (the cylinder ##\mathbb{R}\times S^1##, torus ##S^1 \times S^1## etc.). But how does one define the circle in point set topology? Is it any set homeomorphic to the set ##\left\{(x,y) \in...
  12. C

    Dedekind Cuts & the Real Line: A Countable Set?

    If every Dedekind cut is at a rational it seems that these cuts would only produce a countable set and would not produce the whole real line. So how should I think about it.
  13. C

    Open set (equivalent definitions?)

    I've seen open sets ##S## of a bigger set ##X## being defined as 1) for every ##x\in S## one can find an open disk ##D(x,\epsilon)## centered at ##x## of radius ##\epsilon## such that ##D## is entirely contained in ##S##. Where $$D(x,\epsilon)= \left\{y \in X: d(x,y) < \epsilon\right\}$$...
  14. M

    Two exercises on complex sequences (one about Mandelbrot set)

    Homework Statement . I am trying to solve two exercises about complex sequences: 1) Let ##\alpha \in \mathbb C##, ##|\alpha|<1##. Which is the limit ##\lim_{n \to \infty} \alpha^n##?, do the same for the case ##|\alpha|>1##. 2) Let ##\mathcal M## be the set of the complex numbers ##c## such...
  15. tom.stoer

    The set of the real numbers is closed

    The set of the real numbers is closed. For me this is nearly trivial (*) but perhaps I miss something; a colleagues insists that there are some deeper considerations why this is far from trivial - but I don't get his point (*) A) A set is closed if its complement is open; the complement...
  16. M

    Describe and diagram the set determined by the condition

    Homework Statement 0<\left|x+3\right|<1/4 Homework Equations The Attempt at a Solution (-13/4)<x<(-11/4) and x\neq-3 Thanks in advance. This is my first post and I am unfamiliar with formatting this kind of stuff so I will work on getting better at that aspect.
  17. M

    MHB Calculating the Euler's Totient Function for a Given Integer

    Hey! :o I am looking at an exercise and I got stuck... $n\epsilon \mathbb{N},n>1$ $φ(n)=|\{1 \leq k \leq n :$ the greatest common divisor of $k$ and $n$ is $1\}|$ I am asked to find $φ(n)$,but I don't know how...
  18. G

    Compactness of a set of feasible solutions

    Hi everyone, I am working on a problem in Operations Research but I need to prove a property related to compactness of a set. Although I expect it is quite elementary, I have never studied Analysis at an advanced level so am not sure how to do it. I have an optimisation problem in which a...
  19. R

    Combining loosely correlated data set

    I need some help finding an appropriate statistics model for some experimental data. Thanks in advance for any suggestions. I am trying to compare simulated results from a code that models nuclide concentrations in spent nuclear fuel to experimental data. These concentrations have...
  20. M

    Finding 8 Relations on a Set of 3 Elements with the Same Symmetric Closure

    Homework Statement Show that if a set has 3 elements, then we can find 8 relations on A that all have the same symmetric closure. Homework Equations Symmetric closure ##R^* = R \cup R^{-1} ## The Attempt at a Solution If the symmetric closures of n relations are the same then...
  21. D

    Set Theory Proof: A∩B=Ø implies C∩D=Ø

    Homework Statement Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it...
  22. D

    Set Theory Proof Help: Proving C∩D=Ø When A⊆C and B⊆D

    Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it both ways correct...
  23. M

    Polar coordinates to set up and evaluate double integral

    Homework Statement Use polar coordinates to set up and evaluate the double integral f(x,y) = e-(x2+y2)/2 R: x2+y2≤25, x≥0 The Attempt at a Solution First I just want to make sure I'm understanding this my double integral would be ∫^{\pi/2}_{-\pi/2} because x≥0 ∫^{5}_{0}...
  24. S

    Calculating rate constant from a set of data?

    Homework Statement For a reaction, A + H2O --> B + C We're given that d[A]/dt = k[A]n[H3O+]m And also a table of [A] vs time at T1 and pH 1, pH 2; as well as [A] vs time at T2 and the same pH 1 and 2. From this data, we're to find pseudo-n-order rate constants, and then n itself. Next...
  25. B

    Cardinality of infinite subset of infinite set

    Am a bit confused about the meaning of cardinality. If ## A \subseteq B ##, then is it necessarily the case that ## |A| \leq |B| ##? I am thinking that since ## A \subseteq B ##, an injection from A to B exists, hence its cardinality cannot be greater than that of B? But this cannot be...
  26. M

    Proving a set of functions is bounded in an open set

    Homework Statement . Let ##X## be a complete metric space and consider ##C(X)## the space of continuous functions from ##X## to ##\mathbb R## with the metric ##d_{\infty}##. Suppose that for every ##x \in X##, the set ##\{f(x): f \in C(X)\}## is bounded in ##\mathbb R##. Prove that there exist...
  27. L

    Set of invertible matrices with real entries

    ##GL(n,\mathbb{R})## is set of invertible matrices with real entries. We know that SO(n,\mathbb{R}) \subset O(n,\mathbb{R}) \subset GL(n,\mathbb{R}) is there any specific subgroups of ##GL(n,\mathbb{R})## that is highly important.
  28. K

    MHB Let A and B be two subsets of some universal set. Prove that....

    **Let A and B be two subsets of some universal set. Prove that if $(A\cup B)^c$ = $A^c$ U $B^c$, then A = B.**Attempt: Let $x\in A$. Then $x\in A\cup B$, so $x\notin(A\cup B)^c$. By hypothesis $(A\cup B)^c=A^c\cup B^c$, so $x\notin A^c\cup B^c$. In particular, then, $x\notin B^c$, and therefore...
  29. G

    What is the rigorous definition of set?

    Hi all, first math post here. I was just wondering- after having read from quite a few textbooks that intuitively, a set is a collection of objects- if there's a rigorous definition of the concept of set. It's just out of curiosity- I mean, is a rigorous definition even necessary? I guess I'm...
  30. S

    Selecting a subset from a set such that a given quantity is minimized

    Let A be is a set of some p-dimensional points x \in \mathbb{R}^p. Let d_x^A denote the mean Euclidean distance from the point x to its k nearest points in A (others than x). Let C \subset A be a subset of points chosen randomly from A. We have \Phi(A) = \sum_{x \in A} d_x^C. Now suppose that...
  31. V

    Difference between propositional language and set of all formulas

    I am currently reading Rautenberg's book on mathematical logic, in it he defines a propositional language ##\mathcal{F}##, set theoretically, as the smallest (i.e. the intersection) of all sets of strings ##S## built from propositional variables (##\ p_1,p_2,\ldots##) as well as any binary...
  32. A

    Is P(E) U P(F) Equal to P(E U F)?

    Homework Statement Prove that(power set) P(E) U P(F) is a subset of P(E U F) Homework Equations P(E) U P(F) is a subset of P(E U F) The Attempt at a Solution P(E)U P(F)={x:xεP(E) or xεP(F)} but P(E)={X:X is a subset of E} or P(E)={x:xεX→xεE} so we get P(E)U P(F)={x:xεX→xεE or...
  33. F

    How to set up a double slit experiment?

    I was thinking of finding a new hobby, and thought that playing around with the double slit experiment might be interesting. I was wondering how feasible it would be to set up the double slit experiment, not simply the laser and slit version shown here...
  34. N

    How to calculate a conditionally trancated PDF from an ordered set

    Hello, I am trying to evaluate the following condition: Let x_1 ≥ x_2 ≥ ... ≥ x_L and x_L ≥ y, where y is a fixed deterministic value. What is the conditional PDF of x_j (1 ≤ j ≤ L), given that x_L ≥ y ? (recall that i < L)Note that the conditional PDF of x_j, for the case when x_1 ≥ x_2 ≥...
  35. M

    Set theory: find the intersection

    Homework Statement In a group of 30 people each person twice read a book from books A, B, C. 23 people read book A, 12 read book B and 23 read book C. (a) How many people read books A and B? (b) How many people read books A and C? (c) How many people read books B and C? Homework...
  36. J

    How to prove orthogonality on a set of functions?

    Homework Statement A set of functions, F, is given below. Determine the size of the largest subset of F which is mutually orthogonal on the interval [-1, 1], and find all such subsets of this size. Show all of your work. F = { 1, x, x2 , sin(x), cos(x), cosh(x), sinh(x)}Homework Equations Not...
  37. M

    Family of equicontinuous functions on compact set

    Homework Statement . Let ##X## be a compact metric space. Prove that if ##\mathcal F \subset X## is a family of equicontinuous functions ##f:X \to Y \implies \mathcal F## is uniformly equicontinuous. The attempt at a solution. What I want to prove is that given ##\epsilon>0## there...
  38. V

    Is the empty set always part of the basis of a topology?

    The topology ## T ## on a set ## X ## generated by a basis ## B ## is defined as: T=\{U\subset X:\forall\ x\in U\ there\ is\ a\ \beta\in B:x\in \beta \ and\ \beta\subset U \}. But if ##U## is the empty set, and there has to be a ## \beta ## in ##B## that is contained in ##U##, the empty set...
  39. R

    MHB Approximation property with F sigma and G delta Sets to show a set is measurable

    Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...
  40. K

    MHB Prove that the interval A = [0 , 2) has the same cardinality as the set B = [5 , 6) U [7 , 8)

    Prove that the interval A = [0 , 2) has the same cardinality as the set B = [5 , 6) U [7 , 8) by constructing a bijection between the two sets Attempt: x ↦ x + 5 for x ∈ [0 ; 1) x ↦ x + 6 for x ∈ [1 ; 2) What to do next?
  41. 9

    How to Solve Algebraic Equations with Fractions: (1/n) = (n/100)

    Algebraically, how is this done? I can do it no problem if there is no fraction, but have problems when there is. (1/n) = (n/100)
  42. D

    Set of Commuting Observables for pures states 2p-1,2px and 2s

    Hi, Here I have a question, apparently easy, but that I think it is a bit tricky. Homework Statement Indicate how can a hydrogen atom be prepared in the pure states corresponding to the state vectors ψ2p-1 and ψ2px and ψ2s. It is assumed that spin-related observables are not...
  43. P

    Proving linearly independent set

    1. Prove that if A is symmetric and B is skew-symmetric, then {A,B} is a linearly independent set. I am going to need some help to solve this. Not sure how to begin. Homework Equations The Attempt at a Solution
  44. J

    Span of a Set of Linear Transformations

    How do you show that a set of linear transformations from one vector space to another spans L(V,W)? This isn't a homework question, or even a question that's in the text I'm reading (Friedberg).
  45. A

    Cantor set ℵ , inductive proofs by openly counting.

    I have been looking at the idea of 1:1 correspondence as a method of determining set size/cardinality, and have noticed that the principle allows for inductive proofs, which I think are properly constructed, that can come to conclusions which are clearly wrong under traditional set theory if...
  46. T

    Proving Convexity of a Set: A Proof by Contradiction Approach

    How to show that a set ##C=\{(x,y,z)\in\mathbb{R}^3:x\geq0,z\geq0,xz\geq y^2\}## is convex? I tried a proof by contradiction: Assume that there exist ##c_1=(x_1,y_1,z_1),c_2=(x_2,y_2,z_2)\in C## and ##t\in(0,1)## such that ##tc_1+(1-t)c_2\notin C##. For this to hold, one would have to have...
  47. M

    Showing a set is compact

    Homework Statement Explain why ##f(x,y,z) = x + y - z## must attain both a maximum and a minimum on the sphere ##x^2 + y^2 + z^2 = 81)##. Homework Equations None The Attempt at a Solution I know that any continuous function attains both a maximum and a minimum on a compact set. I defined...
  48. M

    Deriving conclusion from a small data set

    Hi Please see table below GDP growth rates of a group 5 countries. I am trying to derive some conclusions from this small sample. 1. can I conclude that countries in the group have very similar growth rates and there is no significant difference between their growth rates? 2. As...
  49. K

    Set Theory, Functions. Injective/Surjective

    Homework Statement Give f:A→A and g:A→A where f is surjective, g is injective, but f*g is neither surjective nor injecive The Attempt at a Solution I don't know why I can't really think of two... I assume it's easiest to do one in ℝ, but when it comes to producing...
  50. I

    Proving the Truth of 3(b) in Basic Set Theory

    how do I go about doing 3(a) and 3(b)? I'm guessing that for 3(a), it is true, since we have for LHS: P((A or B) and C) we can consider the case P(A and C) by excluding B, and this is a subset of the RHS when we also exclude B: (P(A) and P(C)). We can consider excluding B because...
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