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Propositional Logic CS 1050 (Rosen Section 1.1, 1.2) Proposition A proposition is a statement that is either true or false, but not both. • Atlanta was the site of the 1996 Summer Olympic games. • 1+1 = 2 • 3+1 = 5 • What will my CS1050 grade be? Definition 1. Negation of p Let p be a proposition. The statement “It is Table 1. not the case that p” is The Truth Table for the also a proposition, Negation of a Proposition called the “negation of p ¬p p” or ¬p (read “not p”) p = The sky is blue. T F F T p = It is not the case that the sky is blue. p = The sky is not blue. Definition 2. Conjunction of p and q Let p and q be Table 2. The Truth Table for propositions. The the Conjunction of two propositions proposition “p and q,” denoted by pq is true p q pq when both p and q are true and is false T T T otherwise. This is T F F called the conjunction F T F F F F of p and q. Definition 3. Disjunction of p and q Let p and q be Table 3. The Truth Table for the Disjunction of two propositions. The propositions proposition “p or q,” denoted by pq, is the p q pq proposition that is false T T T when p and q are both T F T false and true otherwise. F T T F F F Definition 4. Exclusive or of p and q Let p and q be Table 4. The Truth Table for the Exclusive OR of two propositions. The propositions exclusive or of p and q, denoted by pq, is the p q pq proposition that is true T T F when exactly one of p T F T and q is true and is F T T false otherwise. F F F Definition 5. Implication pq Let p and q be propositions. Table 5. The Truth Table for The implication pq is the the Implication of pq. proposition that is false when p is true and q is false, and p q pq true otherwise. In this implication p is called the T T T hypothesis (or antecedent or T F F premise) and q is called the F T T conclusion (or F F T consequence). Implications • If p, then q • Not the same as the • p implies q if-then construct • if p,q used in programming • p only if q languages such as • p is sufficient for q If p then S • q if p • q whenever p • q is necessary for p Implications How can both p and q be false, and pq be true? •Think of p as a “contract” and q as its “obligation” that is only carried out if the contract is valid. •Example: “If you make more than $25,000, then you must file a tax return.” This says nothing about someone who makes less than $25,000. So the implication is true no matter what someone making less than $25,000 does. •Another example: p: Bill Gates is poor. q: Pigs can fly. pq is always true because Bill Gates is not poor. Another way of saying the implication is “Pigs can fly whenever Bill Gates is poor” which is true since neither p nor q is true. Related Implications Converse of Contrapositive pq of p q is is the proposition qp q p Definition 6. Biconditional Let p and q be Table 6. The Truth Table for propositions. The the biconditional pq. biconditional pq is the proposition that is true p q pq when p and q have the T T T same truth values and is T F F false otherwise. “p if and F T F only if q, p is necessary F F T and sufficient for q” Practice p: You learn the simple things well. q: The difficult things become easy. • You do not learn the • The difficult things simple things well. p become easy but you • If you learn the simple did not learn the simple things well then the things well. q p difficult things become • You learn the simple easy. pq things well but the • If you do not learn the difficult things did not simple things well, then become easy. the difficult things will p q not become easy. p q Truth Table Puzzle Steve would like to determine the relative salaries of three coworkers using two facts (all salaries are distinct): • If Fred is not the highest paid of the three, then Janice is. • If Janice is not the lowest paid, then Maggie is paid the most. Who is paid the most and who is paid the least? p : Janice is paid the most. •If Fred is not the highest paid q: Maggie is paid the most. of the three, then Janice is. •If Janice is not the lowest paid, r: Fred is paid the most. then Maggie is paid the most. s: Janice is paid the least. p q r s rp s q (rp) (sq) T F F F T F F F T F T F T F F F T T T T T F T F F F T F F F T F T F F Fred, Maggie, Janice p : Janice is paid the most. •If Fred is not the highest paid q: Maggie is paid the most. of the three, then Janice is. •If Janice is the lowest paid, r: Fred is paid the most. then Maggie is paid the most. s: Janice is paid the least. p q r s rp s q (rp) (sq) T F F F T T T F T F T F T F F F T T T F F F T F F F T F F F T F T T T Fred, Janice, Maggie or Janice, Maggie, Fred or Janice, Fred, Maggie Bit Operations A computer bit has two possible values: 0 (false) and 1 (true). A variable is called a Boolean variable is its value is either true or false. Bit operations correspond to the logical connectives: OR AND XOR Information can be represented by bit strings, which are sequences of zeros and ones, and manipulated by operations on the bit strings. Truth tables for the bit operations OR, AND, and XOR 0 1 0 1 0 0 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 0 1 Logical Equivalence • An important technique in proofs is to replace a statement with another statement that is “logically equivalent.” • Tautology: compound proposition that is always true regardless of the truth values of the propositions in it. • Contradiction: Compound proposition that is always false regardless of the truth values of the propositions in it. Logically Equivalent • Compound propositions P and Q are logically equivalent if PQ is a tautology. In other words, P and Q have the same truth values for all combinations of truth values of simple propositions. • This is denoted: PQ (or by P Q) Example: DeMorgans • Prove that (pq) (p q) p q (pq) (pq) p q (p q) TT T F F F F TF T F F T F FT T F T F F FF F T T T T Illustration of De Morgan’s Law (pq) p q Illustration of De Morgan’s Law p p Illustration of De Morgan’s Law q q Illustration of De Morgan’s Law p q p q Example: Distribution Prove that: p (q r) (p q) (p r) p q r qr p(qr) pq pr (pq)(pr) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F Prove: pq(pq) (qp) pq pq pq qp (pq)(qp) TT T T T T TF F F T F FT F T F F FF T T T T We call this biconditional equivalence. List of Logical Equivalences pT p; pF p Identity Laws pT T; pF F Domination Laws pp p; pp p Idempotent Laws (p) p Double Negation Law pq qp; pq qp Commutative Laws (pq) r p (qr); (pq) r p (qr) Associative Laws List of Equivalences p(qr) (pq)(pr) Distribution Laws p(qr) (pq)(pr) (pq)(p q) De Morgan’s Laws (pq)(p q) Miscellaneous p p T Or Tautology p p F And Contradiction (pq) (p q) Implication Equivalence pq(pq) (qp) Biconditional Equivalence

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posted: | 8/28/2011 |

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