近世代数-英文版课件近世代数-英文版课件.pdfVIP

Mathematical Reasoning, Logic and Problem Solving Lecture 4 Dr. Uwe Schauz and Dr. Ignazio Longhi Department of Mathematics, XJTLU Suzhou, PR China October 8th and 9th, 2015 MTH105. Lecture 4 1/25 A little more of logic We defined p ⇒ q to be true whenever p is false. One way of understanding why this is a reasonable definition is to look at ¬(p ⇒ q). The statement “p does not imply q ” means that it is possible that p holds even if q is false. p q p ⇒ q ¬(p ⇒ q) p ∧ ¬q T T T F F T F F T T F T T F F F F T F F The table shows that our definition of p ⇒ q is exactly the one which yields ¬(p ⇒ q) ≡ p ∧ ¬q , as suggested by intuition. MTH105. Lecture 4 2/25 In order to prove that a predicate of the form P (x) ⇒ Q(x) is false, it is enough to produce a counterexample. This is because the negation of P⇒Q is logically equivalent to P∧¬Q: thus P⇒Q is false if and only if we have that both P and ¬Q are true. About CNF and DNF: from the truth table of a compound proposition P, it is easy to find a formula in DNF which is logically equivalent to P. If you want a formula in CNF, you can first express ¬P in DNF and then “push ¬ inside the formula” by the equivalence ¬(a ∧ b) ≡ ¬a ∨ ¬b (so that terms become clauses and DNF becomes CNF). You can find