Question

Show that the statement "For any real numbers a and b, $a^2=b^2$ implies that $a=b$" is not true by giving a counter-example

Answer 1

The given statement can be written in the form of "if-then" as follows
If $a$ and $b$ are real numbers such that $a^2=b^2$ then $a=b$
Let $p:a$ and $b$ are real numbers such that $a^2=b^2$
$q:a=b$
The given statement has to be proved false.
For this purpose it has to be proved that if $p$ then $\sim q$
To show this two real numbers $a$ and $b$ with $a^2=b^2$ are required such that
$a\ne b$
Let $a=1$ and $b=-1$
$a^2={\left(1\right)}^2=1$ and $b^2={(-1)}^2=1$
$\therefore a^2=b^2$
However $a\ne b$
Thus it can be concluded that the given statement is false