Show that the statement “For any real numbers a and b, a^2=b^2 im… - Vidyadip

Question

Show that the statement “For any real numbers a and b, $\text{a}^2=\text{b}^2$ implies that $\text{a = b}$” is not true by giving a counter-example.

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Answer

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

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