PlanetMath: property

(more info)

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property

(Definition)

Let be a set. A property of is a function

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

An element $x\in X$ is said to have or does not have the property

depending on whether $p(x)=\mathit{true}$ or $p(x)=\mathit{false}$ . Any property gives rise in a natural way to the set

$\displaystyle X(p):=\lbrace x\in X \vert \ x$ has property $\displaystyle p\rbrace$

and the corresponding characteristic function $1_{X(p)}$ . The identification of

with $X(p)\subseteq X$ enables us to think of a property of

as a 1-ary, or a unary relation on

. Therefore, one may treat all these notions equivalently.

Usually, a property of can be identified with a so-called propositional function, or a predicate $\varphi(v)$ , where is a variable or a tuple of variables whose values range over . The values of a propositional function is a proposition, which can be interpreted as being either “true” or “false”, so that $X(p)=\lbrace x \mid \varphi(x)$ is $\mathit{true}\rbrace$ .

Below are a few examples:

Let $X=\mathbb{Z}$ . Let $\varphi(v)$ be the propositional function “ is divisible by ”. If is the property identified with $\varphi(v)$ , then $X(p)=3\mathbb{Z}$ .
Again, let $X=\mathbb{Z}$ . Let $\varphi(v_1,v_2):=$ “ is divisible by ” and the corresponding property. Then
$\displaystyle X(p)=\lbrace (m,n)\mid m=np$ , for some $\displaystyle p\in \mathbb{Z}\rbrace,$
which is a subset of $X\times X$ . So is a property of $X\times X$ .
The reflexive property of a binary relation on can be identified with the propositional function $\varphi(V):=\lq\lq \forall a\in X$ , $(a,a)\in V$ ”, and therefore
$\displaystyle X(p)=\lbrace R\subseteq X\times X\mid \varphi(R)$ is $\displaystyle \mathit{true}\rbrace,$
which is a subset of $2^{X\times X}$ . Thus, 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 . Let
$\displaystyle \varphi(\mathcal{V}):=\lq\lq \forall n\in \mathbb{N}, \forall E_1\in \... ...ldots,\forall E_n\in \mathcal{V}, \exists x\in X (x\in E_1\cap \cdots \cap E_n)$ ”
and the corresponding property, then
$\displaystyle X(p)=\lbrace \mathcal{F} \subseteq 2^X\mid \varphi(\mathcal{F})$ is $\displaystyle \mathit{true}\rbrace,$
which is a subset of $2^{2^X}$ . Thus is a property of $2^{2^X}$ .