# property

Let $X$ be a set. A property $p$ of $X$ is a function

 $p\colon X\to\{\mathit{true},\mathit{false}\}.$

An element $x\in X$ is said to have or does not have the property $p$ depending on whether $p(x)=\mathit{true}$ or $p(x)=\mathit{false}$. Any property gives rise in a natural way to the set

 $X(p):=\{x\in X|\ x\text{ has property }p\}$

and the corresponding http://planetmath.org/node/CharacteristicFunctioncharacteristic function    $1_{X(p)}$. The identification of $p$ with $X(p)\subseteq X$ 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 $\varphi(v)$, 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 “true” or “false”, so that $X(p)=\{x\mid\varphi(x)\mbox{ is }\mathit{true}\}$.

Below are a few examples:

• Let $X=\mathbb{Z}$. Let $\varphi(v)$ be the propositional function “$v$ is divisible by $3$”. If $p$ is the property identified with $\varphi(v)$, then $X(p)=3\mathbb{Z}$.

• Again, let $X=\mathbb{Z}$. Let $\varphi(v_{1},v_{2}):=$$v_{1}$ is divisible by $v_{2}$” and $p$ the corresponding property. Then

 $X(p)=\{(m,n)\mid m=np\mbox{, for some }p\in\mathbb{Z}\},$

which is a subset of $X\times X$. So $p$ is a property of $X\times X$.

• The reflexive property of a binary relation on $X$ can be identified with the propositional function $\varphi(V):=\forall a\in X\mbox{, }(a,a)\in V$”, and therefore

 $X(p)=\{R\subseteq X\times X\mid\varphi(R)\mbox{ is }\mathit{true}\},$

which is a subset of $2^{X\times X}$. Thus, $p$ is a property of $2^{X\times X}$.

• In point set topology  , we often encounter the finite intersection property on a family of subsets of a given set $X$. Let

 $\varphi(\mathcal{V}):=\forall n\in\mathbb{N},\forall E_{1}\in\mathcal{V},% \ldots,\forall E_{n}\in\mathcal{V},\exists x\in X(x\in E_{1}\cap\cdots\cap E_{% n})\mbox{''}$

and $p$ the corresponding property, then

 $X(p)=\{\mathcal{F}\subseteq 2^{X}\mid\varphi(\mathcal{F})\mbox{ is }\mathit{% true}\},$

which is a subset of $2^{2^{X}}$. Thus $p$ is a property of $2^{2^{X}}$.

 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