examples of contrapositiveExamplesOfContrapositive2006-11-06 12:56:522006-11-09 21:07:58Examplecontrapositive% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Cl}{\operatorname{Cl}}Recall that the contrapositive of an implication $p \implies q$ is the equivalent implication $\neg q \implies \neg p$, which is read: ``not $q$ implies not $p$''. The following are examples of the contrapositive and converse of a logical statement:
\begin{enumerate}
\item Let $p$ be the statement ``it is raining'' and let $q$ be ``the ground is getting wet''. Then the statement ``if it is raining then the ground is getting wet'' is equivalent to ``if the ground is not getting wet then it is not raining''. Notice that these are both true statements. Notice also that the converse would be ``if the ground is getting wet then it is raining'' (which is not necessarily true!).
\item Let $f:S\to T$ be a function of sets and let $S$ be finite. The contrapositive statement of ``if $f$ is surjective then $T$ is finite'' (a true statement) would be the implication ``if $T$ is not finite then $f$ is not surjective'' (also a true statement). The converse would be ``if $T$ is finite then $f$ is surjective'' (a false statement).
\end{enumerate}