Geometry

A mathematical statement has a truth value within a stated domain. Write P ⇒ Q for “if P, then Q”. This asserts that no case has P true and Q false; it does not assert P. The converse is Q ⇒ P, the inverse is not P ⇒ not Q, and the contrapositive is not Q ⇒ not P. The contrapositive is equivalent to the original implication.

The converse and inverse are equivalent to each other, but need not be equivalent to the original. An equivalence P ⇔ Q requires both directions, often proved separately. Negating P ⇒ Q gives P and not Q, not the inverse.

The quantifier ∀ means “for every”, while ∃ means “there exists”. Always identify the domain: ∀ real x, x² ≥ 0 is true, whereas ∀ real x, x² > 0 fails at zero. Its negation is ∃ real x with x² ≤ 0. Negating an existential statement produces a universal statement with its predicate negated. Several successful examples suggest a conjecture but do not prove a universal claim. A single valid counterexample disproves one.

Check that the counterexample belongs to the specified domain and satisfies the hypothesis.

Proof by contradiction assumes the negation of the desired conclusion and derives an impossibility from established facts. For example, two distinct lines cannot share two distinct points: if they did, both would be the unique line through those two points, contradicting distinctness. State exactly what is contradicted, then reject the assumption. In geometric proofs, separate given facts from conclusions suggested by a sketch. A drawn right angle is not evidence unless it is specified or established.

When communicating a strategy, name the theorem, identify its hypotheses and give the equation it produces.