After learning how to negate an implication, we next compare the related conditional statements formed by reversing or negating its parts. The implication, converse, inverse, and contrapositive are often confused in ordinary language. In mathematical logic, each has a precise symbolic form. In this lesson, converse, inverse, and contrapositive will be distinguished by logical equivalence and truth values. :::definition[Contrapositive] Let $p\to q$ be an implication. The contrapositive of $p\to q$ is $\neg q\to\neg p$. ::: :::definition[Converse] Let $p\to q$ be an implication. The converse of $p\to q$ is $q\to p$. ::: :::definition[Inverse] Let $p\to q$ be an implication. The inverse of $p\to q$ is $\neg p\to\neg q$. ::: Use the sorter to keep the four related conditional forms separate. The original implication keeps the order $p$ then $q$, the converse reverses the order, the inverse negates both parts, and the contrapositive both reverses and negates. The sorter also marks which forms are logically equivalent. This helps prevent the common mistake of treating a statement and its converse as the same claim. :::calculator[Conditional Form Sorter] ::: :::theorem[Contrapositive Equivalence] Let $p$ and $q$ be primitive statements. Then $p\to q\Leftrightarrow\neg q\to\neg p$. ::: :::proof Given that $p$ and $q$ are primitive statements. To prove that $p\to q\Leftrightarrow\neg q\to\neg p$. Using implication as disjunction, $$\begin{equation*} p\to q\Leftrightarrow\neg p\lor q. \end{equation*}$$ Also, $$\begin{align*} \neg q\to\neg p&\Leftrightarrow\neg(\neg q)\lor\neg p\\ &\Leftrightarrow q\lor\neg p\\ &\Leftrightarrow\neg p\lor q. \end{align*}$$ Therefore, both statements are equivalent to $\neg p\lor q$. Hence, $p\to q\Leftrightarrow\neg q\to\neg p$. $\square$ ::: :::theorem[Converse-Inverse Equivalence] Let $p$ and $q$ be primitive statements. Then $q\to p\Leftrightarrow\neg p\to\neg q$. ::: :::proof Given that $p$ and $q$ are primitive statements. To prove that $q\to p\Leftrightarrow\neg p\to\neg q$. The contrapositive of $q\to p$ is $\neg p\to\neg q$. Since an implication is logically equivalent to its contrapositive, $$\begin{equation*} q\to p\Leftrightarrow\neg p\to\neg q. \end{equation*}$$ Hence, the converse and inverse of an implication are logically equivalent to each other. $\square$ ::: :::definition[Non-Equivalence of Implication and Converse] In general, $p\to q\not\Leftrightarrow q\to p$. The truth of an implication does not by itself guarantee the truth of its converse. ::: :::definition[Non-Equivalence of Inverse and Contrapositive] In general, $\neg p\to\neg q\not\Leftrightarrow\neg q\to\neg p$. The inverse is equivalent to the converse, not to the original implication. ::: :::example[Thanksgiving Conditional] Let $p$ mean Today is Thanksgiving, and let $q$ mean Tomorrow is Friday. The implication $p\to q$ says: If today is Thanksgiving, then tomorrow is Friday. The contrapositive $\neg q\to\neg p$ says: If tomorrow is not Friday, then today is not Thanksgiving. The converse $q\to p$ says: If tomorrow is Friday, then today is Thanksgiving. The inverse $\neg p\to\neg q$ says: If today is not Thanksgiving, then tomorrow is not Friday. ::: :::solved-problem Show that an implication is not logically equivalent to its converse. ::: :::solution Let $p\to q$ be an implication. Suppose $p=0$ and $q=1$. Then $p\to q=1$, but $q\to p=0$. Since there is at least one truth assignment for which the two statements have different truth values, $p\to q\not\Leftrightarrow q\to p$. Hence, an implication is not logically equivalent to its converse. ::: :::solved-problem Translate necessary and sufficient conditions using converse and contrapositive. ::: :::solution The statement $p$ is sufficient for $q$ means $p\to q$. The statement $p$ is necessary for $q$ means $q\to p$. Therefore, the converse of $p$ is sufficient for $q$ is $p$ is necessary for $q$. Since $p\to q\Leftrightarrow\neg q\to\neg p$, necessary and sufficient conditions can also be studied through contrapositives. ::: :::exercise For the implication If $n$ is divisible by $4$, then $n$ is even, write the converse, inverse, and contrapositive. ::: :::answer Converse: If $n$ is even, then $n$ is divisible by $4$. Inverse: If $n$ is not divisible by $4$, then $n$ is not even. Contrapositive: If $n$ is not even, then $n$ is not divisible by $4$. ::: :::faq[Frequently Asked Questions] Q: Which statement is equivalent to an implication? A: An implication is logically equivalent to its contrapositive. Q: Is the converse always false? A: No. The converse may be true or false, but it is not guaranteed by the truth of the original implication. Q: Which statement is equivalent to the inverse? A: The inverse is logically equivalent to the converse. ::: :::call-to-action[Compare Conditional Forms] subtitle: Practise writing the four related conditional forms before studying nested conditionals and program logic. button: Next Lesson | /discrete-mathematics/mathematical-logic/logical-equivalence-and-the-laws-of-logic/nested-conditionals-and-program-logic button-ghost: Previous Lesson | /discrete-mathematics/mathematical-logic/logical-equivalence-and-the-laws-of-logic/negation-and-duals-of-conditional-statements :::