property

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

$$p:X\to \{\mathrm{𝑡𝑟𝑢𝑒},\mathrm{𝑓𝑎𝑙𝑠𝑒}\}.$$

An element $x\in X$ is said to have or does not have the property $p$ depending on whether $p(x)=\mathrm{𝑡𝑟𝑢𝑒}$ or $p(x)=\mathrm{𝑓𝑎𝑙𝑠𝑒}$. 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 $\phi (v)$, 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 $X(p)=\{x\mid \phi (x)\text{ is }\mathrm{𝑡𝑟𝑢𝑒}\}$.

Below are a few examples:

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

• Again, let $X=\mathbb{Z}$. Let $\phi (v_1,v_2):=$“$v_1$ is divisible by $v_2$” and $p$ the corresponding property. Then

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

  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 $\phi (V):=\mathrm{`}\mathrm{`}\forall a\in X\text{, }(a,a)\in V$”, and therefore

  $$X(p)=\{R\subseteq X\times X\mid \phi (R)\text{ is }\mathrm{𝑡𝑟𝑢𝑒}\},$$

  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

  $$\phi (𝒱):=\mathrm{`}\mathrm{`}\forall n\in \mathbb{N},\forall E_1\in 𝒱,\mathrm{\dots },\forall E_n\in 𝒱,\exists x\in X(x\in E_1\cap \mathrm{\cdots }\cap E_n)\text{”}$$

  and $p$ the corresponding property, then

  $$X(p)=\{ℱ\subseteq 2^X\mid \phi (ℱ)\text{ is }\mathrm{𝑡𝑟𝑢𝑒}\},$$

  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