Mathematics - Mathematical Reasoning Question with Solution | TestHub

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