Mathematical Statements and Logic

Mathematical Statements

Mathematics relies on a language of absolute precision. Its core building blocks are statements (propositions): declarative assertions that are unambiguously either true or false, never both and never neither.

Simple statements combine into compound ones using logical connectives. Negation ($\neg p$) reverses truth value. Conjunction ($p \land q$) is true only when both inputs are true. Disjunction ($p \lor q$) is true when at least one input is true.

The meaning of any connective is captured by a truth table listing every combination of inputs. A compound of $n$ statements requires $2^n$ rows.

Connectives and Equivalences

Two statements are logically equivalent when they share identical truth tables. This is the bedrock of proof: substituting equivalent statements preserves truth.

$$p \lor q \text{ is true} \iff p \text{ or } q \text{ (or both)}$$

Common Pitfall: Never read mathematical or as the everyday exclusive or. In math, $p \lor q$ is inclusive: it is true when $p$ holds, when $q$ holds, or when both hold. For example, "$x > 0$ or $x < 5$" holds for $x = 3$ because both parts are true.