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Contents Index Dark Mode Prev Up Next \(\newcommand{\im}{{\mathrm{im}}}
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\)
Subsection 1.1.4 Converse, contrapositive, and inverse
Definition 1.1.4.1 .
Let
\(p\) and
\(q\) be propositions, and consider the conditional
\(p \implies q\) . The
converse of
\(p \implies q\) is
\(q \implies p\) . The
contrapositive of
\(p \implies q\) is
\(\lnot q \implies\lnot p\) . The
inverse of
\(p \implies q\) is
\(\lnot p \implies\lnot q\) .
The truth tables for the converse, contrapositive, and inverse of a conditional statement are given in
Figure 1.1.4.3 .
Example 1.1.4.2 . Converse, contrapositive, and inverse.
Consider the conditional “If I studied, then I passed the course.”
The converse is the conditional “If I passed the course, then I studied.”
The contrapositive is the conditional “If I did not pass the course, then I did not study.”
The inverse is the conditional “If I did not study, then I did not pass the course.”
\begin{equation*}
\begin{array}{|*{2}{c|}|*{6}{c|}}
\hline
\amp \amp \amp \amp \text{Conditional} \amp \text{Converse} \amp \text{Contrapositive} \amp
\text{Inverse}\\\hline
p\amp q\amp \lnot p\amp \lnot q\amp p \implies q\amp q \implies p\amp \lnot q
\implies\lnot p\amp \lnot p \implies\lnot q\\
\hline
\true \amp \true \amp \false \amp \false \amp \true \amp \true \amp \true \amp \true \\
\true \amp \false \amp \false \amp \true \amp \false \amp \true \amp \false \amp \true \\
\false \amp \true \amp \true \amp \false \amp \true \amp \false \amp \true \amp \false \\
\false \amp \false \amp \true \amp \true \amp \true \amp \true \amp \true \amp \true \\
\hline
\end{array}
\end{equation*}
Figure 1.1.4.3. Truth tables for converse, contrapositive, and inverse of the conditional \(p\implies q\)
