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.
Homework Statement
Give an example of a set of real numbers whose interior is empty but whose closure is all of the real numbers if it exists. Otherwise, explain why such example cannot be true.
2. The attempt at a solution
For a set S ⊆ X, the closure of S is the intersection of all closed...
Herstein in his book topics of algebra(sec 1.2, 2nd ed) defines A(S) to be set of all one-to-one mappings of S onto itself, S is a non empty set. Is he defining A(S) to be set of all bijections from S-->S or is he defining A(S) to be set of all injections from S--->S.
He uses the word...
Homework Statement
A vessel of cross section 20 cm2 has at the bottom a horizontal capillary tube of length 10 cm and internal diameter 0.5 mm. It is initially filled with water to a height of 50 cm. Find the time taken to empty one half of the content in the vessel. Viscosity of water is...
This is not a homework question. I am trying to figure it out myself.
Let a spherical shell of radius R be filled with a liquid (full). Now let a small hole of area A be made at the bottom of the shell. Find the time required to empty the shell completely.
Does empty space move? What are the laws that govern its motion?
The big-bang set the universe expanding. But universe did not consist of matter at that time. So, what started expanding! Empty space - I suppose. But do the laws of motion govern the motion of empty space as well? Don't we need...
Homework Statement
Much as the title says, I need to construct an example of a linear operator on Hilbert space with empty spectrum. I can very easily construct an example with empty point-spectrum (e.g. the right-shift operator on l_2), but this has very far from empty spectrum.
If I...
Please share the information you have about this topic, post only valid articles and research.
By "blocks of empty space within the known universe" I mean blocks of space that are free of both matter and light i.e. there is no light or matter within that space;empty and dark. And the known...
of loose object in the center. This object is now surrounded by mass, will it hover exactly in the center of the hole, is there no gravity there ? If it is flexible, will it be expanded if it is expandable. How does one calculate this?
(For realism, assume symetrical 2 meter channels from the...
If a sun is giving off photons in every direction in space, does that not mean that all of empty space is full of photons? I know photons have no mass but that still means it is not a perfect vacuum right?
If I take an empty ordinary drinks bottle, place it in my mouth and suck, the sides collapse in wards. Is this because I have reduced the pressure in the bottle and the weight of the atmosphere is crushing it?
Thanks in advance
James
Homework Statement
Empty cubic box of dimension "a" with conducting walls. Two opposite walls are held at a constant potential V, while the other walls are grounded. Find an expression for the electric potential inside the box. (Assume box centered on the origin, walls are all normal to one...
open? A set is open if every point is interior point but empty set has no point! So no?
bounded?no point so cannot measure distance?so no?
perfect? i guess no because there is no point in empty set.
compact?There is no open cover "covering" the empty set right? so no?
Homework Statement
Here is a picture of the problem:
http://img84.imageshack.us/img84/1845/screenshot20100927at111.png
If the link does not work, the problem basically asks:
Let "a" be a non-zero vector in R^n. Let S be the set of all orthogonal vectors to "a" in R^n. I.e., a•x = 0 (where •...
Given any set A, a relation on A is a subset of AxA. Then isn't the empty set a relation also? Doesn't that make it an equivalence relation, vacuously, as well?
I'm asking because in a book there's a problem stating: show there are exactly 5 equivalence relations on a set with 3 elements. I get...
http://www.theglobeandmail.com/news/national/quebec/no-survivors-in-quebec-slide/article1564432/
Thinking of how unexpected and imminent death is, this tragic incident made me ponder for a moment; that when a tragedy hits a family, people pray for at least some survivors in the incident. We...
In ZF (or at least the formulation I'm studying), the empty set axiom (\exists \phi)(\forall z)(\neg (z\in \phi)) is a consequence of the other axioms, namely the axiom of infinity (and separation, though googling says that's not necessary) which is (\exists x) (p) where p is some formula to...
Do all sets include Empty set by default?
Like I have { {a}, {b}, {c} }, its equal to { emptySet, {a}, {b}, {c} }?
See I have this question it goes something like.
The power set of {a, b} is {emptySet, {a}, {b}, {a,b}}
So for all A, B that belongs to the power set {a,b}, A R B where...
There is no such thing as empty space in an atom. String Theory Amendum
Another question that bothers me. We are taught that the atom is mostly empty space and I understand Bohr's atom and the concept of quantum. No problem. However, there is a movement in physics that states the "empty...
Homework Statement
Find the work required to empty a tank in the shape of a hemisphere of radius R meters through an outlet at the top of the tank. The density of water is p kg/m^{3}; the acceleration of a free falling body is g. (Ignore the length of the outlet at the top.)
The attempt...
Problem:
"What is the weight difference between an empty memory stick and the same memory stick when it contains data?"
Additional question:
Can the answer to the previous question be proofed?
I've been wondering two things lately. Why did Einstein make the assumption that the Ricci tensor is 0 in empty space. Is there a physical/mathematical reason? I know later he set it equal to another tensor...which leads to all the cosmological constant business, but I'm just curious why he...
Hi, new guy here.
I once saw a graph on the development of the energy density of the universe, with the X axis measuring time (I believe it was in billion years or so) while the Y axis meansuring the density.
There were three lines in that graph, energy density of empty space, energy...
What is Instance, Empty Constructor ... in Java
How come will return false I put in 2 same name it should return true ?
Teacher p1= new Teacher("Nelson");
System.out.println(p1);
Teacher p2= new Teacher("Nelson");
System.out.println(p2);
System.out.println(p1.equals(p2));
it...
Hi everyone, first time poster.
Considering some of the things discussed here, I apologize if my question sounds simplistic.
If this Earth could be compressed so that all the empty space in atoms would disappear how big would this Earth be?
My science teacher once told me that it would be...
My apologies if the following is a stupid question. But is the empty set disjoint with itself? Certain aspects of set theory has always been counter-intuitive for me.
\oslash\cap\oslash=\oslash?
What is the current measured density contrast inside the cosmological voids of radius about 30 Mpc? By measured I mean somebody constructed a density map based on peculiar velocities map or weak lensing.
I've read a Paper by L. da Costa - Astrophysical Journal 468, L5, 1996 - that constructed...
Hi,
I'm reading HL Royden's real analysis, though my question pertains more to set theory.
Let X be a set. Then the intersection of an empty collection of subsets of X is equal to X. I understand this is not an intersection of empty subsets but it is still very counter-intuitive. Can...
Hi! I'd like to ask the following question.
Does it make sense to take unions and intersections over an empty set?
For instance I came across a definition of a topological space which uses just two axioms:
A topology on a set X is a subset T of the power set of X, which satisfies:
1...
I am beginning to study set theory and came across the following example:
Let \mathcal{A} be the empty family of subsets of \mathbb{R}. Since \mathcal{A} is empty, every member of \mathcal{A} contains all real numbers. That is, ((\forall A)(A\in\mathcal{A}\Rightarrow x\in A)) is true for...
In hypothetical empty space, Bob stays home and Alice flies radially away from home at .9c. In the flat Minkowski coordinates, each determines that the other's clock is running slower than their own. Each determines that the radial distance between them is Lorentz-contracted.
Now consider...
Okay, here’s the concept.
Is it just “empty space” or something else?
But since relativity; we know that space cannot, and does not exist without time,
And it’s a continuum "spacetime" rather than space and time.
So does it means that vacuum is just “empty spacetime”
There arises a...
Hi, I joined recently and I'd like to start a do-discussion about the empty set. Why is it empty and why is there only one empty set?
It's "full of nothing", right?
Apparently the ontology of a singular but empty set is still a hurdle for the philosophers, yet mathematicians use this logical...
These two terms would be the simplest terms in the set theory as they are on the first chapter in my textbook. But why can't i understand it?
In my textbook it says
A relation R in a set is a set of ordered pairs, so any subset of a set of ordered pairs will be a relation. This includes the...
http://uk.reuters.com/article/worldNews/idUKTRE53P2EG20090426
Another virus springs up. Also apparently Texas, California, New York.
More information:
http://online.wsj.com/article/SB124077900486757017.html
http://www.cdc.gov/swineflu/
Homework Statement
We have a tank with some gel and it will be emptied into another empty tank. The air in the first tank wil stay in the first tank. The air in the second tank will stay in the second tank.
What is the equilibrium height of the solution inside the tank.
also, see Excel...
Homework Statement
Use Torricelli's principle to find the time it takes to empty a conical tank of circular cross section standing on its apex whose angle is 45° and has an outlet of cross sectional area 1.0cm². The tank is initially full of water and at time t = 0 the outlet is opened and...
I've thought about this one a few times in the past and can't wrap my head around it, how can space or rather nothing actually curve to cause the effect of gravity?
Is it possible that it just looks like space curves and it's actually caused by something similar to magnetism?
Think about...
Hi guys! I have one question that bothers me for reeeeally long time (and it's NOT homework! :-) I found one question which goes like this "How can physics help you escape from a totally closed empty room?" Of course it means that room doesn't have any doors or windows, it would be too easy to...
What is "empty space"?
I always hear this term: "empty space." It is synonymous with "vacuum," but I am having trouble understanding either of these two ideas.
How can space be "empty?"
I'm following a course in Bohm's theory of QM, and I'm trying to understand what happens to 'empty waves'.
Consider artificial universe: contains 1 H atom only and an external potential which is zero everywhere except at a little barrier.
The H atom particles are traveling along in their...
Homework Statement
If \emptyset has no elements, then \emptyset is open.
The Attempt at a Solution
If \emptyset is closed, then \emptyset has at least an element.
This is a contradiction, so \emptyset must be open.
I am not sure about the validity of my attempt.
At a physics conference on the east coast, feynman says that empty space isn't really empty but empty space make up of tiny elementary particles that have yet to be detected, and he demonstrated that space is not empty using his feynman diagrams. Physicists at the conference rejected feynman's...