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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 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 1ary, 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 socalled 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 $2^{{2^{X}}}$.
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Comments
addition to property
Hi drini,
added a bit to the "property" topic, what do you think?
You already said
> Given any element of a set $X$
I'm just adding why we do it that way.
regards, marijke
http://web.mat.bham.ac.uk/marijke/
Re: addition to property
it's ok, It's not even my object, I only adopted it for fixing some grammar after the original owner negelected it for a month
f
G > H G
p \ /_  ~ f(G)
\ / f ker f
G/ker f