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.
Solution:
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 $a \neq b$.
Let a = 1 and b = –1
$a^{2}=(1)^{2}=1$ and $b^{2}=(-1)^{2}=1$
$\therefore a^{2}=b^{2}$
However, $a \neq b$
Thus, it can be concluded that the given statement is false.