Linearity
$ \mathfrak{Z}\big\{ax[n]+by[n]\big\}=a\mathfrak{Z}\big\{x[n]\big\}+b\mathfrak{Z}\big\{y[n]\big\}\;\;\;\;\;\forall a,b \in \mathbb{C} \;\;\;, \forall signals \; x[n],y[n] $
if
$ \mathfrak{Z}\big\{x[n]\big\} = X(Z)\;with\;ROC = R_1 $ $ \mathfrak{Z}\big\{y[n]\big\} = Y(Z)\;with\;ROC = R_2 $
then
$ ax[n]+bx[n]\longrightarrow aX(Z)+bY(Z)\;\;\;with\;ROC = R_1 \cap R_2 $
Time Shifting
$ \mathfrak{Z}\big\{x[n-n_0]\big\} = z^{-n_0}\mathfrak{Z}\big\{x[n]\big\}\;same\;ROC\;as\;x[n] $
Proof:
let $ n_o $ be a finite integer
$ \mathfrak{Z}\big\{x[n-n_0]\big\}=\sum_{n=-\infty}^{\infty}x[n-n_0]z^{-n}\;\;\;let\;\;\;k = n-n_0 $
= $ \sum_{k=-\infty}^{\infty}x[k]z^{-k-n_0} = z^(-n_0)\sum_{k=-\infty}^{\infty}x[k]z^{-k} $
= $ z^{n_0}X(Z) $
