Propositions and Logical Connectives

Intro to Proofs · Axiom Academy

LESSON Propositions and Logical Connectives The building blocks of mathematical reasoning — statements that are true or false, and the operators that combine them. A proposition is a declarative statement that is either true (T) or false (F), but not both. Questions, commands, and statements that depend on an unknown variable are not propositions — they have no definite truth value. "2 + 2 = 4" (true) · "5 is even" (false) · "Paris is in France" (true) "What time is it?" (question) · "x + 3 = 7" (depends on x) · "Close the door!" (command) We combine propositions using three fundamental operators: NOT (negation), AND (conjunction), and OR (disjunction). Watch the inputs cycle through all four combinations and each operator re-evaluate. 3. The Conditional Statement (→) The most important — and often most confusing — operator is the conditional or implication : p → q, read "if p, then q." Think of it as a promise: "If it rains, I'll bring an umbrella." The promise is broken only if it rains and I don't. Terminology: in p → q we call p the hypothesis (or antecedent), and q the conclusion (or consequent). 4. The Biconditional Statement (↔) The biconditional p ↔ q means "p if and only if q" (often abbreviated "iff"). It is true exactly when both propositions have the same truth value — the animation lights up only when P and Q agree. "A number is even ↔ it is divisible by 2." Both directions hold: even numbers are divisible by 2, and numbers divisible by 2 are even.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.