Question

The statement $p$ : For any real numbers $x,y$ if $x=y$, then $2x+a=2y+a$ when $a\in Z$

• A. is true
• B. is false
• C. its contrapositive is not true
• D. None of these

Answer 1

Answer: (A)

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