Logic and Quantifiers

Real Analysis · Axiom Academy

The precise language of proof: how connectives build statements, and how quantifiers let one claim speak for an entire domain. 1. Connectives Combine Truth Values A proposition is a statement with a definite truth value, T or F. A connective is a machine: feed it the truth values of P and Q , and it returns one output. Watch the inputs flow through each operator below — the output it lights up is fixed entirely by the inputs. Four connectives carry almost all of mathematics. Each is defined completely by its truth table — the output for every combination of inputs: Implication rewritten with and — same truth table The biconditional is true exactly when P and Q agree 2. Quantifiers Range Over a Domain A connective combines two fixed statements. A quantifier does something stronger: it makes one claim about every element of a domain. There are two. Watch the scan sweep the domain — the same elements, judged by two different rules. Universal: P(x) holds for every x Existential: P(x) holds for at least one x True only if every element passes. A single failing element makes it false. Example: — every natural number is non-negative. True. True if at least one element passes. Finding one witness settles it. Example: — a witness is . True.

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