The statement p: For any real numbers x, y if x=y , then 2 x+a=2...

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 \mathbb{Z} \end{array} Contrapositive method: The contrapositive statement of $p$ is "For any real numbers $x, y$ if 2 \mathrm{x}+\mathrm{a} \neq2 \mathrm{y}+\mathrm{a}, where \mathrm{a} \in \mathrm{Z}, then \mathrm{x} \neq \mathrm{y}." Given, 2 x+a ≠ 2 y+a$$\begin{array}{l} \Rightarrow2 x ≠ 2 y \\ \Rightarrow x ≠ y \end{array}$$ Hence, the given statement is true.$