We prove the statement $p$ is true by contrapositive method and by direct method.

Direct method: For any real number $x$ and $y$,

$\begin{array}{l}

x=y \\

\Rightarrow 2 x=2 y \\

\Rightarrow 2 x+a=2 y+a \text { a for some } a \in Z

\end{array}$

Contrapositive method: The contrapositive statement of $p$ is "For any real numbers $x, y$ if $2 \mathrm{x}+\mathrm{a} \neq 2 \mathrm{y}+\mathrm{a}$, where $\mathrm{a} \in \mathrm{Z}$, then $\mathrm{x} \neq \mathrm{y}$."

Given, $2 x+a \neq 2 y+a$

$\begin{array}{l}

\Rightarrow 2 x \neq 2 y \\

\Rightarrow x \neq y

\end{array}$

Hence, the given statement is true.