Mathematical Reasoning - Implications, Converse, and Contrapositive

An implication or conditional statement is a compound statement formed by connecting two simple statements $p$ and $q$ with the words 'if' and 'then'. It is denoted as $p \implies q$, where $p$ is called the antecedent (hypothesis) and $q$ is the consequent (conclusion). Visually, this can be represented as a directional arrow or a flow chart pointing from the cause $p$ to the effect $q$.

The converse of a conditional statement $p \implies q$ is formed by interchanging the hypothesis and the conclusion, resulting in $q \implies p$. It is important to note that the converse is not logically equivalent to the original statement; imagine a Venn diagram where circle $P$ is inside circle $Q$; while everything in $P$ is in $Q$, not everything in $Q$ is necessarily in $P$.

The contrapositive of $p \implies q$ is formed by negating both the hypothesis and the conclusion and then interchanging them, denoted as $\sim q \implies \sim p$. A fundamental rule of logic is that a conditional statement is always logically equivalent to its contrapositive. Visually, if $P$ is a subset of $Q$, then anything outside of $Q$ must also be outside of $P$.

The inverse of $p \implies q$ is formed by negating both the hypothesis and the conclusion without switching their positions, denoted as $\sim p \implies \sim q$. Like the converse, the inverse is not logically equivalent to the original implication, but it is logically equivalent to the converse of the original statement.

A biconditional statement, written as $p \iff q$ (p if and only if q), is true only when both $p \implies q$ and $q \implies p$ are true. This can be visualized as two overlapping circles in a Venn diagram that are perfectly coincident, meaning $p$ and $q$ represent the exact same logical conditions.

The truth table for an implication $p \implies q$ shows that the statement is only 'False' when the antecedent $p$ is true and the consequent $q$ is false. In all other scenarios, including when $p$ is false, the implication is considered 'True'.

The negation of an implication $p \implies q$ is logically equivalent to $p \land \sim q$. This means the only way to disprove the rule 'If $p$, then $q$' is to find a case where $p$ happens, but $q$ does not happen.