Postulates and Axioms

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 L...

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

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 L...

BASE AXIOMS AND SUBBASE AXIOMS IN M-FUZZIFYING CONVEX SPACES

Based on a completely distributive lattice $M$, base axioms and subbase axioms are introduced in $M$-fuzzifying convex spaces. It is shown that a mapping $mathscr{B}$ (resp. $varphi$) with the base axioms (resp. subbase axioms) can induce a unique $M$-fuzzifying convex structure with  $mathscr{B}$ (resp. $varphi$) as its base (resp. subbase). As applications, it is proved that bases and subbase...

Definitions , Axioms , Postulates , Propositions , and Theorems from Euclidean and Non - Euclidean Geometries , 4 th Ed by Marvin Jay

Behavior postulates and corollaries--1949.