Empty Definition and 271 Threads

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.
In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set.

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  1. B

    Is the Empty Set a Vector Space?

    In the book it states that the span of the empty set is the trivial set because a linear combination of no vectors is said to be the 0 vector. I really don't know how they came up with at? Is it just defined to be like that? After doing some research, I figured that since the empty set is a...
  2. M

    Can you reach absolute zero in complete empty space

    Ok, so I know that the laws of physics say reaching absolute zero temperature is impossible, but suppose we took a box that was perfectly insulated in completely empy space, and I took all the particles out of it to create a vacuum. Now, since there are no particles in the box, then wouldn't...
  3. V

    Showing two epsilon balls intersection is empty

    Homework Statement Suppose x,y \in X which is a normed linear space and x\neq y . Prove that \exists r>0 such that B(x,r) \cap B(y,r)=∅ Homework Equations Epsilon Ball B(x,r)={z \in X:||x-z||<r} The Attempt at a Solution So my attempt here is via contradiction and its not...
  4. X

    Are Empty Sets Equal? Exploring the Concept of Equality in Mathematics

    I want to ask that, are you empty sets equal ?
  5. F

    Time it takes for a water tank to empty

    Homework Statement A cylindrical tank of diameter 2R contains water to a depth d. A small hole of diameter 2r is opened in the bottom of the tank. r<<R, so the tank drains slowly. Find an expression for the time it takes to drain the tank completely.Homework Equations...
  6. N

    Assembly how to check empty string?

    I am asking user to enter a number if user hits enter key instead of entering a value then i need to exit the program. see code below ; i put the value into eax cmp AL,0 ;i compare is al reg is empty je exitProgram ; some instructions exitProgram: Invoke ExitProcess,0 Public...
  7. A

    What is an empty family of subsets?

    At the beginning of a question my book is saying: "Let the universe of discourse be the set ℝ of real numbers, and let A be the empty family of subsets of ℝ.." How on Earth can an empty family contain sets which are subsets of something? A family is a set whose elements are sets. If the...
  8. A

    Proof that empty set is subset of every set

    My book presents the theorem that the empty set is a subset of every set as follows: Let A be a set. Let x be any object. Assume x belongs to empty set. Then x belongs to A. Thus the empty set is a subset of A. In other words, the proposition [(x belongs to empty set) implies (x...
  9. Z

    Is more than 99.9% of any object is just empty space?

    Because of the proportionately huge spaces between the wave-particles in the atom, we accept that 99.9% of any object is empty space. If I understand this at all, the 'empty space' to which this refers is nothing, i.e. not a thing. Of course we would like to describe or quantify 'empty...
  10. B

    Understanding Proper Subsets: An Explanation of the Empty Set in Set Theory

    Ok I am a bit confused here. According to the definition of a proper subset means that everything in set A is in set b and a set always contains an extra pair of brackets. But in this example C={∅,{∅}} why is this correct ∅ ⊆C instead of {∅} ⊆C for the first object?
  11. B

    Are the members of (x,y) if and only if (y = x + 3) & (y = x – 3) an empty set?

    Is the set of all integers where <x,y> \in R if and only if (y = x + 3) & (y = x – 3) an empty set?
  12. Z

    Adobe - Warning Empty Page - Read Out Loud

    I have tried listening to books in PDF format using the Adobe 'Read Out Loud' function in their PDF reader. It works without problem on one book, though on a second book the voice states "Warning! Empty page." for every page in the book. Would anyone have any ideas as to what might be...
  13. Q

    Will the expanding universe really end up cold and empty?

    If you take it to its extremes and space continues to expand at an accelerated rate, what happens when? 1) space is expanding so fast that virtual particles cannot annihilate. 2) if the space between quarks ends up expanding then the energy needed to pull them apart will spawn new quarks...
  14. A

    Union and Intersection of empty class of sets

    why intersection of empty class of sets is the whole space while their union is null set? Book writes that an element will fail to be in the intersection if it fails to be in one of the sets of the class but since there is nothing in the empty class so there is nothing in the empty class that...
  15. W

    Sum of Ideals with Empty Intersection

    Homework Statement If a ring R contains two ideals B and C with B+C=R and B\capC=0, prove that B and C are rings and R\congB x C. Homework Equations B+C={all b+c|b\inB and c\inC} The Attempt at a Solution So far I've discovered that if the unit of R is in one of the ideals then...
  16. B

    What happens to an empty can in space?

    This may have been asked before but I am new here and my kid came to me and asked me this and I don't have an answer so the question is you take an empty can unsealed (nondescript container) into space expose the container to the vacuum of space seal the container and bring it back to earth...
  17. Useful nucleus

    What is the definition of the empty intersection?

    I read that an empty collection of sets, denote it by λ, is a little problematic when one considers \bigcup_{A\inλ}A and \cap_{A\inλ}A. I can see that the union should be ∅. However, for the intersection it was argued that if one considers a set X to be the universe of the discussion then the...
  18. S

    Fortran Gfortran, finding empty text lines?

    Win 7 MinGW compilers. I have a txt file that has empty lines that I'd like to pass by when reading them. Each line is assumed to be 85 characters. Here's a snippet of code: Open(unit=astro_in, file="METEOR_LegacyInputToNML.txt", status = "OLD") do while(.True.)...
  19. S

    What is the meaning of ∅ used in the context of a relation?

    Hi. I am reading Halmos's Naive Set Theory book. I have the following doubts. 1. In chapter 7. on relations, he says "The least exciting relation is the empty one. (To prove that ∅ is a set of ordered pairs, look for an element of ∅ that is not an ordered pair) ". Whenever someone talks...
  20. D

    Prove that if S is a subset of A then S is an empty set.

    Homework Statement Let S be a set such that for each set A, we have S\subseteqA. Show that S is an empty set. 3. Relevant equations 10.6 Proposition. For each set A, we have empty set \subseteq A. The Attempt at a Solution Solution. Consider any set A and a set S such that S\subseteqA. Choose...
  21. S

    Is the matter wave of a photon affected by the presence of an electron?

    how empty (or filled) is the empty (matter) wave? The de broglie-bohm interpretation posits that: in a single particle double slit experiment, the photon goes through one of the slits/paths and the matter waves go through both the slits. the path without the photon and just the matter wave...
  22. H

    Forces Acting on a Rod in Empty Space: Motion & Kinetic Energy

    Suppose I have a metal rod in empty space with a uniform mass distribution. Suppose a force acts perpendicular to the rod at some distance R from the rod's centre of mass over some time Δt. There are no other forces acting on the rod. (1) What type of motion will the rod have over Δt? Will...
  23. A

    Atom Empty Space: What is in Between Electrons and Nuclei?

    I've always heard that atoms are 99.9999996% empty space, now I am wondering is the empty space between the orbiting electrons and the nuclei just empty; a vacuum...?? My question is: What is in the empty space of the atom?(for I doubt it is actually purely empty) What is in the space between...
  24. J

    The notion of injectivity is undefined on the empty set function?

    When defining the notion of injectivity of functions it uses notation, for example for a function f, f(a_{1})=f(a_{2}) where a_{1},a_{2} are in the domain of the function. Since the empty set function, i.e., \emptyset \subseteq \emptyset \times A for some set A, has the empty domain, the...
  25. J

    Is every empty set function continuous?

    Can I use the definition of continuity of function from Baby Rudin, setting X as empty set? Rudin does not specify X is a non-empty set but he supposes p is in X. Anyway if I use it for empty set X, then is a function with a domain E which is a subset of X continuous at p? One more...
  26. J

    Is the Empty Set Considered a Metric Space?

    Homework Statement Is empty set a metric space? Homework Equations None. The Attempt at a Solution It seems so because all the metric properties are vacuously satisfied. Mabe the question had better be put like this: Does mathematicians tend to think empty set as a metric space...
  27. A

    Is there empty space with null energy?

    hi my question is simple: is there empty space with null energy? thx.
  28. teroenza

    (0,0) Empty Interval? Basic Set Theory

    Homework Statement Give an example of an indexed family of sets such that the intersection of any finite subfamily is not empty, but the intersection when the index=infinity, is empty. The Attempt at a Solution The family I came up with is the exclusive interval (-1/k , 1/k) where...
  29. S

    Two Planets Orbiting an Empty Spot

    Okay so I just had this problem on my astronomy midterm and I was sort of stumped by it. I hope I can remember all of it. Basically there are two planets orbiting around what appears to be a blank area of space. From observations we can see that they are not in a binary orbit with each other...
  30. C

    Proving the Empty Intersection of Intervals using Natural Numbers

    Homework Statement Prove that \bigcap_{n=0}^{\inf} (0,\frac{1}{n})=\emptyset The Attempt at a Solution since 0 is not included in our interval. eventually I will get to (0,0) because I could pick a real as close to zero as I wanted and there would be a natural such that...
  31. J

    What Is the Power Set of the Empty Set?

    Homework Statement Is this true or false Homework Equations {{ø}} is a subset of P({ø,{ø}}) The Attempt at a Solution I'm thinking that it is since to make the subset {ø} into a power set it becomes {{ø}}?
  32. A

    Is the Empty Set Axiom Redundant in Ultrafilters?

    One of the four defining axioms of an ultrafilter is that it doesn't contain the empty set (according to Wikipedia, and a talk I was listening to today). Isn't this implied by the other axioms? If an ultrafilter U on X contained the empty set, then it also contains every superset, including...
  33. S

    Solve Spivak's Rocket in Empty Space

    I'm reading Spivak's mechanics book and I'm stumped on some of the math in his derivation of an analytic expression for the motion of a rocket in empty space. Let \mathbf{v}(t) be the rockets velocity, let \mathbf{q}(t) be the velocity at which fuel is ejected from the rocket, and let m(t) be...
  34. Z

    Defining an Empty Set Metric Space: Understanding Properties of d

    Can we define a metric space (\emptyset, d)? The metric is the part that confuses me, since it seems like all of the required properties of d are satisfied since they are "not not satisfied", but I'm not sure. Thank you!
  35. G

    Proving that U∩A is Empty iff U∩Cl(A) is Empty

    Let U be an open subset of a metric space X,and A be an arbitrary subset of X. Prove that the intersection of U and closure of A is empty if and only if the intersection of U and A be empty.
  36. V

    Mercury rotating the earth in an empty region of the galaxy

    1. A: Homework Statement If the Earth and Mercury happened to be pulled out of the solar system by the gravitational force of a passing neutron star, and then left in orbit about each other in an otherwise empty region of the galaxy, our lovely planet would probably get very cold. But...
  37. S

    Dark Energy, Dark Matter, and Empty Space

    Dark Energy, Dark Matter, and "Empty Space" I'm new to this forum, and I've looked around but did not find this discussed. Apologies if I missed it. Has anyone in the research of Astrophysics considered the "missing mass" of the Universe to be contained in the structure of space-time itself...
  38. W

    Is the conduction band of metals empty at absolute zero?

    Hi, I understand for metals the conduction band is usually partially filled. However, at absolute zero, is the conduction band empty (for metals)? I'm asking this because as far as I know, the definition of valence band is "the highest occupied band at absolute zero" However, I have...
  39. J

    Zero-point energy in empty space

    Could the zero-point field energy at each point in empty space be zero because it is canceled by its own negative gravitational self-energy? Inside particles zero-point energy modes that are larger than the particle size are excluded leading to an excess negative gravitational energy inside...
  40. 1

    Fortran Fortran 77 help making an empty array (or blank list if they exist in fortran)

    I need to create an empty array but the length of the array is determined by the start and end numbers entered by the user for example if the range of numbers the user was interested in was 4-10 inclusive then i want the array to be of length 7 How to I go about doing this? Other...
  41. A

    Exploring Empty Functions in Real Analysis

    I'm currently reading Sterling Berberian's Foundations of Real Analysis, and the first chapter had an overview of foundational mathematics from axiomatic set theory to constructive proof of the real numbers. I was looking over this chapter, and I found this exercise in the functions section...
  42. N

    Empty spaces and virtual particles in atom

    as i know,bohr's discovery reveals that atom consist of central nucleus and electron orbit around them in empty spaces my question is-why aren't virtual particles being produced in those empty spaces?
  43. N

    Electromagnetic fields and empty space

    is it possible for electromagnetic field to exist without matter?
  44. L

    Proving something is not the empty set

    Homework Statement for every n in the natural numbers, Vn is a nonempty closed subset of a compact space T. Vn+1 is contained in Vn. Prove the Intersection of all the Vn's from n=1 to infinity does not equal the empty set. Homework Equations The Attempt at a Solution This question...
  45. J

    How do electrons and protons interact in atoms?

    The most abundant element in the universe is hydrogen. The hydrogen atom, like all atoms, is composed of a nucleus orbited by a(n) electron(s). The most likely orbit of the electron in a hydrogen atom, according to the Bohr model, is about 5.3 X 10-11 meters. The volume of the hydrogen atom then...
  46. J

    Does the empty Milne Universe have a horizon problem?

    I have a question about the horizon size of the empty Milne Universe. The empty Milne Universe expands linearly with time. This implies that the horizon size for the Milne Universe is given by: d_H = Integral [ t=t_early to t_0 ] ( dt / t ) d_H = log t_0 - log t_early Thus the...
  47. A

    How Can Redshift be Expressed as a Function of DM in an Empty Universe Model?

    Hi there! So basically I'm trying to get the function which expresses redshift as a function of DM in the empty universe model (Omega=0). I've downloaded the type 1A supernova data from http://www.astro.ucla.edu/~wright/sne_cosmology.html, now I'd like to plot the Omega=0 curve. My GR...
  48. S

    How to deal with a probability conditioned on empty

    Hi everyone, The question: Pr( A|\Phi )=? where \Phi means empty Is it equal to 1 or not? what is the meaning of this probability? thanks.
  49. C

    Unpacking the Mystery of the Empty Set

    If the empty set has no elements then how can it have subsets. Or are we just saying because they don't have anything in them there are equal. If a set is equal to its subset then why is it a subset. And How many subsets does the empty set have.
  50. J

    Calculating Luminosity Distance in an Empty Cosmology

    If had empty cosmology? What would happen if we had no matter or radiation or dark energyand only had curvature? How would this effect luminosity distance? From the friedman equations I get that H2=-k/R2. Is this right?
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