Disproving a Statement By Counter Example
Proofs are often hard. It is often easier to prove that a statement is not true by finding a counterexample.Example: Is it rue that if
\[x \gt y\]
then \[x^2 \gt y^2\]
?No it is not. Take
\[x=1, \: y=-2\]
then \[1 \gt -2 \rightarrow x \gt y\]
but \[1^2=1 \lt (-2)^2=4 \rightarrow x^2 \lt y^2\]
.Is it true that if
\[x \gt y\]
then \[\frac{1}{x} \lt \frac{1}{y}\]
?No it is not. Take
\[x=2, \: y=-1\]
. Then \[x \gt y \]
but \[\frac{1}{2}= \frac{1}{x} \gt \frac{1}{y}=- \frac{1}{2}\]