Definitions , Axioms , Postulates , Propositions , and Theorems from Euclidean and Non - Euclidean Geometries

Logic Rule 0 No unstated assumptions may be used in a proof. Logic Rule 1 Allowable justifications. 1. “By hypothesis . . . ”. 2. “By axiom . . . ”. 3. “By theorem . . . ” (previously proved). 4. “By definition . . . ”. 5. “By step . . . ” (a previous step in the argument). 6. “By rule . . . ” of logic. Logic Rule 2 Proof by contradiction (RAA argument). Logic Rule 3 The tautology ∼ (∼ S)⇐⇒ S Logic Rule 4 The tautology ∼ (H =⇒ C)⇐⇒ H ∧ (∼ C). Logic Rule 5 The tautology ∼ (S1 ∧ S2)⇐⇒ (∼ S1∨ ∼ S2). Logic Rule 6 The statement ∼ (∀xS(x)) means the same as ∃x(∼ S(x)). Logic Rule 7 The statement ∼ (∃xS(x)) means the same as ∀x(∼ S(x)). Logic Rule 8 The tautology ((P =⇒ Q) ∧ P ) =⇒ Q. Logic Rule 9 The tautologies 1. ((P =⇒ Q) ∧ (Q =⇒ R) =⇒ (P =⇒ R). 2. (P ∧Q) =⇒ P and (P ∧Q) =⇒ Q. 3. (∼ Q =⇒∼ P ) =⇒ ((P =⇒ Q). Logic Rule 10 The tautology P =⇒ (P∨ ∼ P ). Logic Rule 11 (Proof by Cases) If C can be deduced from each of S1, S2, · · · , Sn individually, then (S1∨S2∨· · ·Sn) =⇒ C is a tautology. Logic Rule 12 Euclid’s “Common Notions” 1. ∀X (X = X) 2. ∀X ∀Y (X = Y ⇐⇒ Y = X) 3. ∀X ∀Y ∀X ((X = Y ∧ Y = Z) =⇒ X = Z) 4. If X = Y and S(X) is a statement about X, then S(X)⇐⇒ S(Y ) Undefined Terms: Point, Line, Incident, Between, Congruent. Basic Definitions 1. Three or more points are collinear if there exists a line incident with all of them. 2. Three or more lines are concurrent if there is a point incident with all of the them. 3. Two lines are parallel if they are distinct and no point is incident with both of them. 4. {←→ AB} is the set of points incident with ←→ AB. Incidence Axioms: IA1: For every two distinct points there exists a unique line incident on them. IA2: For every line there exist at least two points incident on it. IA3: There exist three distinct points such that no line is incident on all three.