property
Let X be a set. A property p of X is a function
p:Xβ{π‘ππ’π,ππππ π}. |
An element xβX is said to have or does not have the property p depending on whether p(x)=π‘ππ’π or p(x)=ππππ π. Any property gives rise in a natural way to the set
X(p):={xβX|x has property p} |
and the corresponding http://planetmath.org/node/CharacteristicFunctioncharacteristic function 1X(p). The identification of p with X(p)βX enables us to think of a property of X as a 1-ary, or a unary relation on X. Therefore, one may treat all these notions equivalently.
Usually, a property p of X can be identified with a so-called propositional function, or a predicate Ο(v), where v is a variable or a tuple of variables whose values range over X. The values of a propositional function is a proposition, which can be interpreted as being either βtrueβ or βfalseβ, so that X(p)={xβ£Ο(x) is π‘ππ’π}.
Below are a few examples:
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β’
Let X=β€. Let Ο(v) be the propositional function βv is divisible by 3β. If p is the property identified with Ο(v), then X(p)=3β€.
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β’
Again, let X=β€. Let Ο(v1,v2):=βv1 is divisible by v2β and p the corresponding property. Then
X(p)={(m,n)β£m=np, for some pββ€}, which is a subset of XΓX. So p is a property of XΓX.
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β’
The reflexive property of a binary relation on X can be identified with the propositional function Ο(V):=``βaβX, (a,a)βVβ, and therefore
X(p)={RβXΓXβ£Ο(R) is π‘ππ’π}, which is a subset of 2XΓX. Thus, p is a property of 2XΓX.
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β’
In point set topology
, we often encounter the finite intersection property on a family of subsets of a given set X. Let
Ο(π±):=``βnββ,βE1βπ±,β¦,βEnβπ±,βxβX(xβE1β©β―β©En)β and p the corresponding property, then
X(p)={β±β2Xβ£Ο(β±) is π‘ππ’π}, which is a subset of 22X. Thus p is a property of 22X.
Title | property |
Canonical name | Property |
Date of creation | 2013-03-22 14:01:29 |
Last modified on | 2013-03-22 14:01:29 |
Owner | drini (3) |
Last modified by | drini (3) |
Numerical id | 15 |
Author | drini (3) |
Entry type | Definition |
Classification | msc 00A05 |
Synonym | attribute |
Synonym | propositional function |
Related topic | Subset |
Related topic | CharacteristicFunction |
Related topic | Relation |
Related topic | ClosureOfARelationWithRespectToAProperty |
Defines | unary relation |
Defines | predicate |