Its unit set is $~ {\cal SG}_{n,0} = \{ Id_{A} ~|~ \emptyset \neq A
\subseteq \{ x_{1}, x_{2},\ldots, x_{n}\} \}$. The cardinals of these finite sets are given by:

$~~~~~~~~~~~~~~~|~{\cal SG}_{n}~|~=~\sum\limits_{k=1}^{n} k!  {n
\choose k},~~~~~ |~{\cal SG}_{n,0}~| ~=~2^{n} - 1.$\
[*(ii) *]{} [*The Galois groupoid $ {\cal G}al( {\cal E}/ K ) $*]{}. Let $ F / K $ be an extension field of a field $K$, i.e. $K$ is a subfield of $F$. We consider an indexed family $ {\cal E}:=
(E_{i})_{i\in I} $ of intermediate fields $ E_{i} $, that is $ K
\subseteq E_{i}
\subseteq F $ for each $ i\in I.$ Let\
$\Gamma:= {\cal G}al({\cal E}/K) = \{ \varphi : E_{i} \to E_{i}~|~ \varphi~ \hbox{is a K -automorphism } \}
 ~$ and\
$~ \Gamma_{0}:= {\cal G}al({\cal E}/K)_{0} = \{ Id_{E_{I}}~|~ i\in I\} $.

Let $ \Gamma_{(2)}:= \{(\varphi ,\psi)\in \Gamma \times \Gamma | D(\varphi)=D(\psi) \} $. The structure functions $~ {\overline}{\alpha}, {\overline}{\beta} : \Gamma \to
\Gamma_{0},~ {\overline}{\iota} : \Gamma \to \Gamma~ $ and $~ {\overline}{m} :
\Gamma_{(2)} \to \Gamma ~$ are given by:\
$ {\overline}{\alpha}(\varphi):= Id_{D(\varphi)},~~ {\overline}{\beta}(\varphi):=
Id_{D(\varphi)},~~ {\overline}{\iota}(\varphi):=
\varphi^{-1}~~ $ and $~~ {\overline}{m}(\varphi,\psi):= \varphi \circ \psi $.\
Then $ {\cal G}al({\cal E}/K) $ is a groupoid 