Mathematical Reasoning - Implications (If-then, If and only if)

Conditional Statement (Implication): A statement formed by joining two simple statements $p$ and $q$ with the connective 'if...then' is called a conditional statement, denoted by $p \rightarrow q$. In this structure, $p$ is called the antecedent (or hypothesis) and $q$ is the consequent (or conclusion). Visually, this represents a directed flow of logic from one condition to its result.

Truth Values of $p \rightarrow q$: The implication $p \rightarrow q$ is considered False only when the antecedent $p$ is True and the consequent $q$ is False. In all other cases—when $p$ is False (regardless of $q$) or when both are True—the statement is True. Imagine a promise: the promise is only broken if the condition is met but the result is not delivered.

Converse, Inverse, and Contrapositive: For any conditional statement $p \rightarrow q$, three related statements can be formed: the Converse ($q \rightarrow p$), the Inverse ($\sim p \rightarrow \sim q$), and the Contrapositive ($\sim q \rightarrow \sim p$). It is a fundamental rule of logic that a statement is always logically equivalent to its contrapositive, which can be visualized as two different ways of looking at the same logical boundary.

Biconditional Statement (If and only if): A biconditional statement $p \leftrightarrow q$ is the conjunction of two implications: $(p \rightarrow q) \wedge (q \rightarrow p)$. It is True only when both $p$ and $q$ have the same truth value (both True or both False). Visually, this is represented by a double-headed arrow, indicating that $p$ and $q$ are mutually dependent and equivalent in truth.

Negation of an Implication: The negation of $p \rightarrow q$ is logically equivalent to $p \wedge \sim q$. This means that the opposite of 'If $p$, then $q$' is ' $p$ is true AND $q$ is false'. Note that the negation of an 'if-then' statement is not another 'if-then' statement, but a conjunction using 'and'.

Equivalent Expressions: The implication $p \rightarrow q$ can be expressed in several ways in English: '$p$ is sufficient for $q$', '$q$ is necessary for $p$', '$p$ only if $q$', and '$q$ if $p$'. Understanding these linguistic variations is key to translating word problems into logical symbols.