Plane Geometry - Bruce E. Shapiro

20 SECTION 5. LOGIC AND PROOFour theorems in the following manner:If [hypothesis] then [conclusion]We can give our statements names, like A and B, in which case we write:A ⇒ Bwhich we read as “If A then B” or “A implies B.”For a theorem to be accepted as a true, it must have a proof. A is alist of statements that justifies a theorem. Each step must be justified (orexplained) by one of the following methods:ˆ By hypothesis ... (assume that ...)ˆ By axiom X ... (or theorem, definition, postulate, ...)ˆ By step Y ... (an earlier step in the proof)ˆ By a rule of symbolic logicWe will discuss some of the rules of symbolic (formal) logic shortly.There are two special types of theorems: a lemma, and a corollary. Logicallythere is no difference between a theorem, a lemma, and a corollary. Alemma is a theorem which is not really interesting (according to the author)in itself, or is a result that is not pertinent to the subject at hand, but isonly stated only because it makes the proof of some other theorem moreinteresting. A corollary is a theorem that follows almost immediately as aresult of another theorem with very little proof.A statement A in our logical system will only be allowed to have two values:True and False (we may denote these values by T and F).Just writing a statement does not make it true:If △ABC is any triangle then it is equilateralwould have a truth value of False.The value of any implication (A ⇒ B) is given by the following truthtable:A B A ⇒ Btrue true truetrue false falsefalse true truefalse false trueThe negation operation (∼ A) turns true to false and false to true. We usenegation to prove theorems according to method of RAA (Reductio ad« CC BY-NC-ND 3.0. Revised: 18 Nov 2012