Discrete Mathematics, Chapter : Propositional Logic
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1 Discrete Mathematics, Chapter : Propositional Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
2 Outline 1 Propositions 2 Logical Equivalences 3 Normal Forms Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
3 Propositions A proposition is a declarative sentence that is either true or false. Examples of propositions: The Moon is made of green cheese. Trenton is the capital of New Jersey. Toronto is the capital of Canada = = 2 Examples that are not propositions. Sit down! What time is it? x + 1 = 2 x + y = z Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
4 Propositional Logic Constructing Propositions Propositional Variables: p, q, r, s,... The proposition that is always true is denoted by T and the proposition that is always false is denoted by F. Compound Propositions; constructed from logical connectives and other propositions Negation Conjunction Disjunction Implication Biconditional Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
5 Disjunction The disjunction of propositions p and q is denoted by p q and has this truth table: p q p q T T T T F T F T T F F F Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
6 Conjunction The disjunction of propositions p and q is denoted by p q and has this truth table: p q p q T T T T F F F T F F F F Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
7 Implication If p and q are propositions, then p q is a conditional statement or implication which is read as if p, then q and has this truth table: p q p q T T T T F F F T T F F T In p q, p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). Implication can be expressed by disjunction and negation: p q p q Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
8 Understanding Implication In p q there does not need to be any connection between the antecedent or the consequent. The meaning depends only on the truth values of p and q. This implication is perfectly fine, but would not be used in ordinary English. If the moon is made of green cheese, then I have more money than Bill Gates. One way to view the logical conditional is to think of an obligation or contract. If I am elected, then I will lower taxes. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
9 Different Ways of Expressing p q if p, then q if p, q q unless p q if p p is sufficient for q q is necessary for p a sufficient condition for q is p p implies q p only if q q when p q whenever p q follows from p a necessary condition for p is q Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
10 Converse, Contrapositive, and Inverse q p is the converse of p q q p is the contrapositive of p q p q is the inverse of p q Example: Find the converse, inverse, and contrapositive of It is raining is a sufficient condition for my not going to town. Solution: converse: If I do not go to town, then it is raining. inverse: If it is not raining, then I will go to town. contrapositive: If I go to town, then it is not raining. How do the converse, contrapositive, and inverse relate to p q? Clicker 1 converse contrapositive? 2 converse inverse? 3 contrapositive inverse? Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
11 Biconditional If p and q are propositions, then the biconditional proposition p q has this truth table p q p q T T T T F F F T F F F T p q also reads as p if and only if q p iff q. p is necessary and sufficient for q if p then q, and conversely p implies q, and vice-versa Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
12 Precedence of Logical Operators Thus p q r is equivalent to (p q) r. If the intended meaning is p (q r) then parentheses must be used. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
13 Satisfiability, Tautology, Contradiction A proposition is satisfiable, if its truth table contains true at least once. Example: p q. a tautology, if it is always true. Example: p p. a contradiction, if it always false. Example: p p. a contingency, if it is neither a tautology nor a contradiction. Example: p. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
14 Logical Equivalence Definition Two compound propositions p and q are logically equivalent if the columns in a truth table giving their truth values agree. This is written as p q. It is easy to show: Fact p q if and only if p q is a tautology. Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
15 De Morgan s Laws (p q) p q (p q) p q Truth table proving De Morgan s second law. p q p q (p q) (p q) p q T T F F T F F T F F T T F F F T T F T F F F F T T F T T Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
16 Important Logical Equivalences Domination laws: Identity laws: Idempotent laws: Double negation law: Negation laws: p T T, p F F p T p, p F p p p p, p p p ( p) p p p T, p p F The first of the Negation laws is also called law of excluded middle. Latin: tertium non datur. Commutative laws: p q q p, p q q p Associative laws: (p q) r p (q r) (p q) r p (q r) Distributive laws: p (q r) (p q) (p r) p (q r) (p q) (p r) Absorption laws: p (p q) p, p (p q) p Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
17 More Logical Equivalences Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
18 A Proof in Propositional Logic To prove: (p ( p q)) p q (p ( p q)) p ( p q) by De Morgan s 2nd law p ( ( p) q) by De Morgan s first law p (p q) by the double negation law ( p p) ( p q) by the 2nd distributive law F ( p q) because p p F ( p q) F by commutativity of disj. p q by the identity law for F Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
19 Conjunctive and Disjunctive Normal Form A literal is either a propositional variable, or the negation of one. Examples: p, p. A clause is a disjunction of literals. Example: p q r. A formula in conjunctive normal form (CNF) is a conjunction of clauses. Example: (p q r) ( p r) Similarly, one defines formulae in disjunctive normal form (DNF) by swapping the words conjunction and disjunction in the definitions above. Example: ( p q r) ( q r) (p r). Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
20 Transformation into Conjunctive Normal Form Fact For every propositional formula one can construct an equivalent one in conjunctive normal form. 1 Express all other operators by conjunction, disjunction and negation. 2 Push negations inward by De Morgan s laws and the double negation law until negations appear only in literals. 3 Use the commutative, associative and distributive laws to obtain the correct form. 4 Simplify with domination, identity, idempotent, and negation laws. (A similar construction can be done to transform formulae into disjunctive normal form.) Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
21 Example: Transformation into CNF Transform the following formula into CNF. (p q) (r p) 1 Express implication by disjunction and negation. ( p q) ( r p) 2 Push negation inwards by De Morgan s laws and double negation. (p q) ( r p) 3 Convert to CNF by associative and distributive laws. (p r p) ( q r p) 4 Optionally simplify by commutative and idempotent laws. (p r) ( q r p) and by commutative and absorbtion laws (p r) Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter / 21
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