Plane Geometry - Bruce E. Shapiro

SECTION 5. LOGIC AND PROOF 21absurdum): To prove that A ⇒ B, assume ∼ B and deduce somethingthat is not true.We define the operations of and (A∧B) and or (A∨B) as meaning follows:A B A ∧ B A ∨ BT T T TT F F TF T F TF F F FWe can remember this truth table by the following rule: A∨B means eitherA is true or B is true, or both are true; and A ∧ B means both A and Bare true.The Law of Excluded Middles says that for any statement A, either Ais true, or ̸ A is true:∀P, P ∨ ∼ PThe converse of the theorem A ⇒ B is B ⇒ A. The converse is a completelydifferent statement, and may or may not be true.If both a theorem and its converse are true, we call the theorem a logicalequivalence. We write this as A ⇐⇒ B or “A iff B” is read as “A if andonly if B” and means(A ⇒ B) ∧ (B ⇒ A)The contrapositive of a theorem A ⇒ B is ∼ B ⇒∼ A. The contrapositiveis equivalent to the original theorem. This is demonstrated by thefollowing truth table.A B A ⇒ B ∼ A ∼ B ∼ A ⇒∼ BT T T F F TT F F F T FF T T T F TF F T T T TThe universal quantifier ∀x is read as “for all x.” The statement (∀x)(S(x))means “for all x, the statement S(x) is true.”The existential quantifier ∃y is read as “there exists y.”Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.