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 $$X(p):=\lbrace x\in X | \ x\text{ has property }p\rbrace$$ and the corresponding characteristic 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 $X(p)=\lbrace x \mid \varphi(x)\mbox{ is }\mathit{true}\rbrace$ .

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 $$X(p)=\lbrace (m,n)\mid m=np\mbox{, for some }p\in \mathbb{Z}\rbrace,$$ 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 $$X(p)=\lbrace R\subseteq X\times X\mid \varphi(R)\mbox{ is }\mathit{true}\rbrace,$$ 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 $$X(p)=\lbrace \mathcal{F} \subseteq 2^X\mid \varphi(\mathcal{F})\mbox{ is }\mathit{true}\rbrace,$$ which is a subset of $2^{2^X}$ . Thus $p$ is a property of $2^{2^X}$ .