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Q.
The statement $p$ : For any real numbers $x , y$ if $x = y$, then $2 x + a =2 y + a$ when $a \in Z$
Mathematical Reasoning
Solution:
We prove the statement $p$ is true by contrapositive method and by direct method.
Direct method: For any real number $x$ and $y$.
$x=y \Rightarrow 2 x=2 y $
$\Rightarrow 2 x+a=2 y+a$ for some $a \in Z$
Contrapositive method: The contrapositive statement of $p$ is "For any real numbers $x , y$ if $2 x + a \neq 2 y + a$,
where $a \in Z$, then $x \neq y$ "
Given, $2 x+a \neq 2 y+a$
$\Rightarrow 2 x \neq 2 y$
$ \Rightarrow x \neq y$
Hence, the given statement is true.