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1 Gachon University Chulyun Kim
2 2 Outline Propositional Logic Propositional Equivalences Predicates and Quantifiers Nested Quantifiers Rules of Inference Introduction to Proofs
3 3 1.1 Propositional Logic Proposition Logic (1) The rules of logic Design of computer circuits Construction of computer programs Verification of the correctness of programs
4 4 Propositional Logic (2) Proposition Declarative sentence True or false, but not both Examples Toronto is the capital of Canada = 2 What time is it?
5 5 Logical Operators (1) Propositional variables Variables that represent propositions p, q, r, s Negation: p (or p) p: Today is Friday p: Today is not Friday Conjunction: p q p: Today is Friday q: It is raining today p q: Today is Friday and it is raining today p p T F F T p q p q T T T T F F F T F F F F
6 6 Logical Operators (2) Disjunction: p q p: Today is Friday q: It is raining today p q: Today is Friday or it is raining today Exclusive: p q True when exactly one of p and q is true and is false otherwise p q p q T T T T F T F T T F F F p q p q T T F T F T F T T F F F
7 7 Conditional Statements If p, then q p q p: hypothesis( or antecedent) q: conclusion( or consequence) Converse: q p Inverse: p q Contrapositive: q p A conditional statement and its contrapositive are equivalent If it is raining, then the home team wins p q p q T T T T F F F T T F F T Converse: If the home team wins, then it is raining Inverse: If it is not raining, then the home team does not win Contrapositve: If the home team does not win, then it is not raining
8 8 Biconditional Statement p if and only if q p q p iff q p: You can take the flight q: You buy a ticket p q: You can take the flight if and only if you buy a ticket p q p q T T T T F F F T F F F T
9 9 Precedence of Operators Precedence of logical operators ( p) q is the same as p q (p q) r is the same as p q r Truth table of (p q) (p q) Operator Precedence () p q q p q p q (p q) (p q) T T F T T T T F T T F F F T F F F T F F T T F F
10 10 Consistence of Rules If the rules are not consistent, there is no way to satisfy all rules together { p q, p, p q } { p q, p, p q } p q p p q p q p q (p q) p (p q) (p q) p (p q) T T F T T T F F T F F T F F F F F T T T F T T F F F T F F T F F
11 11 Logic and Bit Operations Bit A symbol with two possible values, 0 and 1 Bit string Truth Value Sequence of zero or more bits Length of a string is the number of bits in the string Bitwise OR/AND/XOR Bit T 1 F bitwise OR bitwise AND bitwise XOR
12 Tautology and Contradiction Tautology A compound proposition that is always true Contradiction A compound proposition that is always false Contingency A compound proposition that is neither a tautology nor a contradiction p p p p p p T F T F F T T F Propositional Equivalences
13 13 Logical Equivalences (1) p q : compound propositions p and q are called logically equivalent if p q is a tautology De Morgan s Laws (p q) p q (p q) p q p q p q p q p p q p q T T F T T T F F F F F T T T T F F T T T
14 14 Logical Equivalences (2) Equivalence p T p, p F p p T T, p F F p p p, p p p ( p) p p q q p, p q q p Name Identity laws Domination laws Idempotent laws Double negation law Commutative laws (p q) r p (q r), (p q) r p (q r) Associative laws p (q r) (p q) (p r), p (q r) (p q) (p r) (p q) p q, (p q) p q p (p q) p, p (p q) p p p T, p p F Distributive laws De Morgan s laws Absorption laws Negation laws
15 15 Constructing New Logical Equivalences Show that (p q) and p q are logically equivalent (p q) ( p q) ( p) q by De Morgan law p q by double negation law Show that (p q) (p q) is a tautology (p q) (p q) (p q) (p q) ( p q) (p q) by De Morgan law ( p p) ( q q) by associative and commutative laws T T by negation law T by domination law
16 Predicates x is greater than 3 x: variable is greater than 3: predicate P P(x) Let Q(x,y) denote the statement x=y+3 Q(1,2) is false but Q(3,0) is true n-place predicate (or n-ary predicate) P(x 1, x 2,, x n ) Predicates and Quantifiers
17 17 Quantifiers Universal quantification: x P(x) A predicate is true for every element If P(x) is x<2, is x P(x) true for the domain of integers? Existential quantification: x P(x) A predicate is true for one or more elements If P(x) is x<2, is x P(x) true for the domain of integers? Precedence of Quantifiers Quantifiers and have higher precedence then all logical operators x P(x) Q(x) means ( x P(x)) Q(x)
18 18 Logical Equivalences Involving Quantifiers Statement x(p(x) Q(x)) x P(x) xp(x) Equivalent Statement xp(x) xq(x) x P(x) x P(x) There is an honest politician Every politician is dishonest All Americans eat cheeseburgers Some American does not eat cheeseburgers
19 Nested Quantifiers x y(x+y=0) x Q(x) where Q(x) is y(x+y=0) For every real number x, there is a real number y such that x+y=0 y x(x+y=0) y Q(y) where Q(y) is x(x+y=0) There is a real number y such that for every real number x, x+y=0 Negating nested quantifiers x y(xy=1) x y(xy=1) x y (xy=1) x y(xy 1) Nested Quantifiers
20 Valid Arguments Argument Sequence of propositions (p 1, p 2,, p n ) Premises (p 1, p 2,, p n-1 ) All but the final proposition in the argument Conclusion (p n ) The final proposition The argument is valid if the truth of all its premises implies that the conclusion is true Tautology: (p 1 p 2 p n-1 ) p n If you have a password, then you can log onto the network You have a password Therefore, You can log onto the network 20 premises conclusion 1.6 Rules of Inference
21 21 Rules of Inference Rule of Inference Name p p q q q p q p p q q r p r p q p q Modus ponens Modus tollens Hypothetical syllogism Disjunctive syllogism Rule of Inference Name p p q p q p p q p q p q p r q r Addition Simplification Conjunction Resolution
22 22 Build Arguments (1) Hypotheses It is not sunny this afternoon and it is colder than yesterday We will go swimming only if it is sunny If we do not go swimming, then we will take a canoe trip If we take a canoe trip, then we will be home by sunset Conclusion We will be home by sunset 1. p q Hypothesis 2. p Simplification (1) 3. r p Hypothesis 4. r Modus tollens (2) and (3) 5. r s Hypothesis 6. s Modus ponens (4) and (5) 7. s t Hypothesis 8. t Modus ponens using (6) and (7) p: it is sunny this afternoon q: it is colder than yesterday r: we will go swimming s: we will take a canoe trip t: we will be home by sunset
23 23 Build Arguments (2) Hypotheses Jasmine is skiing or it is not snowing It is snowing or Bart is playing hockey Conclusion Jasmine is skiing or Bart is playing hockey 1. p q Hypothesis 2. p r Hypothesis 3. q r Resolution (1) and (2) p: it is snowing q: Jasmine is skiing r: Bart is playing hokey
24 24 Build Arguments (3) Show that the hypotheses (p q) r and r s imply the conclusion p s 1. (p q) r Hypothesis 2. (p r) (p q) Distributive law 3. p r Simplification 4. r s Hypothesis 5. r s 6. p s Resolution (3) and (5)
25 25 Normal Forms Disjunctive Normal Form (DNF) A formula which is equivalent to a given formula and consists of a sum of elementary products (p q) q ( p q) (q q) Conjunctive Normal Form (CNF) A formula which is equivalent to a given formula and consists of a product of elementary sums (p q) q ( p q) q 1.7 Normal Forms
26 26 Principal Disjunctive Normal Form Minterm Each variable occurs either negated or nonnegated in a conjuction If there are n variables, there will be 2 n minterms Principal disjunctive normal form An equivalent formula consisting of disjunctions of minterms Sum of products canonical form Example p q = (p q) (p q) ( p q)
27 27 Principal Conjunctive Normal Form Maxterm Each variable occurs either negated or nonnegated in a disjuction If there are n variables, there will be 2 n maxterms Principal disjunctive normal form An equivalent formula consisting of conjunctions of maxterms Product of sums canonical form Example p q = ( p q) (p q)
28 28 Terminology Theorem Statement that can be shown to be true Axiom Statement we assume to be true Example: For every real number x, x+0=x Lemma A less important theorem that is helpful in the proof of other results Corollary A theorem that can be established directly from a theorem Conjecture A statement that is being proposed to be true 1.8 Introduction to Proof
29 29 Methods of Proving Theorems Direct proof Proof by contraposition Proof by contradiction
30 30 Direct Proof First, we assume that the hypothesis is true and show that the conclusion is true Definition n is odd(or even) if there exists an integer k such that n=2k+1(or 2k) Theorem If n is an odd integer, then n 2 is odd Proof By the definition of an odd integer, it follows that n=2k+1, where k is some integer Then, n 2 = (2k+1) 2 = 4k 2 +4k+1 = 2(2k 2 +2k)+1 By the definition of an odd integer, n 2 is odd
31 31 Proof by Contraposition Since a theorem is equivalent to its contraposition, the theorem is true if the contraposition is true Theorem If 3n+2 is odd, then n is odd Contraposition If n is even, 3n+2 is even Proof First assume that n is even n = 2k Next show 3n+2 is even 3n+2 = 3(2k)+2 = 6k+2 = 2(3k+1) By the definition of an even integer, 3n+2 is even
32 32 Proof by Contradiction We want to prove that a statement p is true If we can find a contradiction q such that p q is true, p must be false since q is always false It means that p is true Show that at least four of any 22 days must fall on the same day of the week p: there are at most three of any 22 days fall on the same day of the week Supposing that p is true, there can be at most 21 days and this contradicts the hypothesis that we have 22 days Consequently, we know that p is true
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