Presburger arithmetic

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.

In mathematics, the power set (or powerset) of any set S, written ${\mathcal {P}}(S)$ , P(S), ℙ(S), ℘(S) or 2^S, is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.^[1]

Any subset of ${\mathcal {P}}(S)$ is called a family of sets over S.

Example

If S is the set {x, y, z}, then the subsets of S are:

{} (also denoted $\varnothing$ , the empty set)
{x}
{y}
{z}
{x, y}
{x, z}
{y, z}
{x, y, z}

and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.^[2]

Properties

If S is a finite set with |S| = n elements, then the number of subsets of S is $|{\mathcal {P}}(S)|=2^{n}$ . This fact, which is the motivation for the notation 2^S, may be demonstrated simply as follows,

We write any subset of S in the format

\{\omega _{1},\omega _{2},\ldots ,\omega _{n}\}

where

\omega _{i},1\leq i\leq n

, can take the value of

0

or

1

. If

\omega _{i}=1

, the

i

-th element of S is in the subset while the

i

-th element is not in the subset otherwise. Clearly the number of distinct subsets that can be constructed this way is

2^{n}

.

Cantor's diagonal argument shows that the power set of a set (whether infinite or not) always has strictly higher cardinality than the set itself (informally the power set must be larger than the original set). In particular, Cantor's theorem shows that the power set of a countably infinite set is uncountably infinite. For example, the power set of the set of natural numbers can be put in a one-to-one correspondence with the set of real numbers (see cardinality of the continuum).

The power set of a set S, together with the operations of union, intersection and complement can be viewed as the prototypical example of a Boolean algebra. In fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras this is no longer true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra (see Stone's representation theorem).

The power set of a set S forms an Abelian group when considered with the operation of symmetric difference (with the empty set as the identity element and each set being its own inverse) and a commutative monoid when considered with the operation of intersection. It can hence be shown (by proving the distributive laws) that the power set considered together with both of these operations forms a Boolean ring.

Representing subsets as functions

In set theory, X^Y is the set of all functions from Y to X. As "2" can be defined as {0,1} (see natural number), 2^S (i.e., {0,1}^S) is the set of all functions from S to {0,1}. By identifying a function in 2^S with the corresponding preimage of 1, we see that there is a bijection between 2^S and ${\mathcal {P}}(S)$ , where each function is the characteristic function of the subset in ${\mathcal {P}}(S)$ with which it is identified. Hence 2^S and ${\mathcal {P}}(S)$ could be considered identical set-theoretically. (Thus there are two distinct notational motivations for denoting the power set by 2^S: the fact that this function-representation of subsets makes it a special case of the X^Y notation and the property, mentioned above, that |2^S| = 2^|S|.)

This notion can be applied to the example above in which $S=\{x,y,z\}$ to see the isomorphism with the binary numbers from 0 to 2ⁿ−1 with n being the number of elements in the set. In S, a 1 in the position corresponding to the location in the set indicates the presence of the element. So {x, y} = 110.

For the whole power set of S we get:

{ } = 000 (Binary) = 0 (Decimal)
{x} = 100 = 4
{y} = 010 = 2
{z} = 001 = 1
{x, y} = 110 = 6
{x, z} = 101 = 5
{y, z} = 011 = 3
{x, y, z} = 111 = 7

Relation to binomial theorem

The power set is closely related to the binomial theorem. The number of sets with $k$ elements in the power set of a set with $n$ elements will be a combination $C(n,k),$ also called a binomial coefficient.

For example the power set of a set with three elements, has:

$C(3,0)=1$ set with 0 elements
$C(3,1)=3$ sets with 1 element
$C(3,2)=3$ sets with 2 elements
$C(3,3)=1$ set with 3 elements.

Algorithms

If $S\!$ is a finite set, there is a recursive algorithm to calculate ${\mathcal {P}}(S)$ .

Define the operation ${\mathcal {F}}(e,T)=\{X\cup \{e\}|X\in T\}$

In English, return the set with the element $e\!$ added to each set $X\!$ in $T\!$ .

If $S=\{\}\!$ ,then ${\mathcal {P}}(S)=\{\{\}\}$ is returned.
Otherwise:

Let $e\!$ be any single element of $S\!$ .
Let $T=S\setminus \{e\}\!$ , where ' $S\setminus \{e\}\!$ ' denotes the relative complement of $\{e\}\!$ in $S\!$ .
And the result: ${\mathcal {P}}(S)={\mathcal {P}}(T)\cup {\mathcal {F}}(e,{\mathcal {P}}(T))$ is returned.

In other words, the power set of the empty set is the set containing the empty set and the power set of any other set is all the subsets of the set containing some specific element and all the subsets of the set not containing that specific element.

There are other more efficient ways to calculate the power set. For example, use a list of the n elements of S to fix a mapping from the bit positions of n-bit numbers to those elements; then with a simple loop run through all the 2ⁿ numbers representable with n bits, and for each contribute the subset of S corresponding to the bits that are set (to 1) in the number. When n exceeds the word-length of the computer (typically 64 in modern CPUs) the representation is naturally extended by using an array of words instead of a single word.

Subsets of limited cardinality

The set of subsets of S of cardinality less than κ is denoted by ${\mathcal {P}}_{\kappa }(S)$ or ${\mathcal {P}}_{<\kappa }(S)\,.$ Similarly, the set of non-empty subsets of S might be denoted by ${\mathcal {P}}_{\geq 1}(S)\,.$

Power object

A set can be regarded as an algebra having no nontrivial operations or defining equations. From this perspective the idea of the power set of X as the set of subsets of X generalizes naturally to the subalgebras of an algebraic structure or algebra.

Now the power set of a set, when ordered by inclusion, is always a complete atomic Boolean algebra, and every complete atomic Boolean algebra arises as the lattice of all subsets of some set. The generalization to arbitrary algebras is that the set of subalgebras of an algebra, again ordered by inclusion, is always an algebraic lattice, and every algebraic lattice arises as the lattice of subalgebras of some algebra. So in that regard subalgebras behave analogously to subsets.

However there are two important properties of subsets that do not carry over to subalgebras in general. First, although the subsets of a set form a set (as well as a lattice), in some classes it may not be possible to organize the subalgebras of an algebra as itself an algebra in that class, although they can always be organized as a lattice. Secondly, whereas the subsets of a set are in bijection with the functions from that set to the set {0,1} = 2, there is no guarantee that a class of algebras contains an algebra that can play the role of 2 in this way.

Certain classes of algebras enjoy both of these properties. The first property is more common, the case of having both is relatively rare. One class that does have both is that of multigraphs. Given two multigraphs G and H, a homomorphism h: G → H consists of two functions, one mapping vertices to vertices and the other mapping edges to edges. The set H^G of homomorphisms from G to H can then be organized as the graph whose vertices and edges are respectively the vertex and edge functions appearing in that set. Furthermore the subgraphs of a multigraph G are in bijection with the graph homomorphisms from G to the multigraph Ω definable as the complete directed graph on two vertices (hence four edges, namely two self-loops and two more edges forming a cycle) augmented with a fifth edge, namely a second self-loop at one of the vertices. We can therefore organize the subgraphs of G as the multigraph Ω^G, called the power object of G.

What is special about a multigraph as an algebra is that its operations are unary. A multigraph has two sorts of elements forming a set V of vertices and E of edges, and has two unary operations s,t: E → V giving the source (start) and target (end) vertices of each edge. An algebra all of whose operations are unary is called a presheaf. Every class of presheaves contains a presheaf Ω that plays the role for subalgebras that 2 plays for subsets. Such a class is a special case of the more general notion of elementary topos as a category that is closed (and moreover cartesian closed) and has an object Ω, called a subobject classifier. Although the term "power object" is sometimes used synonymously with exponential object Y^X, in topos theory Y is required to be Ω.

Functors and quantifiers

In category theory and the theory of elementary topoi, the universal quantifier can be understood as the right adjoint of a functor between power sets, the inverse image functor of a function between sets; likewise, the existential quantifier is the left adjoint.^[3]

Notes

↑ Devlin (1979) p.50
↑ Puntambekar (2007), Template:Google books quote
↑ Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 See page 58

References

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

Property Brokers and Team Managers – Looking for good Actual Estate Agency to join or contemplating which is the Finest Property Agency to join in Singapore? Join Leon Low in OrangeTee Singapore! In OrangeTee, we've much more attractive commission structure than before, enrichment courses, 10 most vital components to hitch OrangeTee and 1 motive to join Leon Low and his Workforce. 1. Conducive working environment

Via PropNex International, we continually construct on our fame in the international property enviornment. Click here for more of our abroad initiatives. Instances have modified. We don't see those unlawful hawkers anymore. Instead, nicely dressed property brokers were seen reaching out to people visiting the market in the morning. Real estate can be a lonely enterprise and it is straightforward to really feel demoralised, especially when there are no enquiries despite your greatest effort in advertising your shopper's property. That is the place having the fitting assist from fellow associates is essential. Our firm offers administration services for condominiums and apartments. With a crew of qualified folks, we assist to make your estate a nicer place to stay in. HDB Flat for Hire 2 Rooms

Achievers are all the time the first to check new technologies & providers that can help them enhance their sales. When property guru first began, many brokers didn't consider in it until they began listening to other colleagues getting unbelievable outcomes. Most brokers needs to see proof first, before they dare to take the first step in attempting. These are often the late comers or late adopters. There is a purpose why top achievers are heading the wave or heading the best way. Just because they try new properties in singapore issues ahead of others. The rest just observe after!

Firstly, a Fraudulent Misrepresentation is one that is made knowingly by the Representor that it was false or if it was made without belief in its fact or made recklessly without concerning whether or not it is true or false. For instance estate agent A told the potential consumers that the tenure of a landed property they are considering is freehold when it is really one with a ninety nine-yr leasehold! A is responsible of constructing a fraudulent misrepresentation if he is aware of that the tenure is the truth is a ninety nine-yr leasehold instead of it being freehold or he didn't consider that the tenure of the house was freehold or he had made the assertion with out caring whether or not the tenure of the topic property is in fact freehold.

I such as you to be, am a brand new projects specialist. You've got the conception that new tasks personnel should be showflat certain. Should you're eager, let me train you the right way to master the entire show flats island vast as a substitute of getting to stay just at 1 place. Is that attainable you may ask, well, I've achieved it in 6 months, you can too. Which company is well-recognized and is actually dedicated for developing rookie within the industry in venture sales market with success? Can a rookie join the company's core group from day one? I wish to propose a third class, which I have been grooming my agents in the direction of, and that is as a Huttons agent, you will be able to market and have knowledge of ALL Huttons projects, and if essential, projects exterior of Huttons as properly.

GPS has assembled a high workforce of personnel who are additionally well-known figures in the native actual property scene to pioneer this up-and-coming organization. At GPS Alliance, WE LEAD THE WAY! Many people have asked me how I managed to earn S$114,000 from my sales job (my third job) at age 24. The reply is easy. After graduation from NUS with a Historical past diploma, my first job was in actual estate. Within the ultimate part of this series, I interview one of the top agents in ERA Horizon Group and share with you the secrets to his success! Learn it RIGHT HERE

Notice that the application must be submitted by the appointed Key Government Officer (KEO) such as the CEO, COO, or MD. Once the KEO has submitted the mandatory paperwork and assuming all documents are in order, an email notification shall be sent stating that the applying is permitted. No hardcopy of the license might be issued. A delicate-copy could be downloaded and printed by logging into the CEA website. It takes roughly four-6 weeks to course of an utility.

22 year-old Systems Analyst Rave from Merrickville-Wolford, has lots of hobbies and interests including quick cars, property developers in singapore and baking. Always loves visiting spots like Historic Monuments Zone of Querétaro.

Here is my web site - cottagehillchurch.com
Template:Planetmath reference
Template:Nlab
Template:Nlab

Template:Logic Template:Set theory

[1] Devlin (1979) p.50

[2] Puntambekar (2007), Template:Google books quote

[3] Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 See page 58

[1]

[2]

[3]

Presburger arithmetic

Contents

Example

Properties

Representing subsets as functions

Relation to binomial theorem

Algorithms

Subsets of limited cardinality

Power object

Functors and quantifiers

Notes

References

External links

Navigation menu

Presburger arithmetic

Example

Properties

Representing subsets as functions

Relation to binomial theorem

Algorithms

Subsets of limited cardinality

Power object

Functors and quantifiers

Notes

References

External links

Navigation menu

Search