Normal Forms Note: all ppts about normal forms are skipped.

Transcription

1 Normal Forms Note: all ppts about normal forms are skipped. Well formed formula (wff) also called formula, is a string consists of propositional variables, connectives, and parenthesis used in the proper manner. E.g. ((p q) ( p r)) p q r is a disjunction expression, and p q r is a conjunction expression. Product for conjunction, sum for disjunction. An elementary product is a conjunction of literals. An elementary sum is a disjunction of literals. A variable of the negation of a variable is called a literal. 1

2 Disjunction/Conjunction Normal Form Disjunctive normal form (DNF) a formula which is equivalent to a given formula and consists of a sum of elementary products E.g. (p q) q ( p q) (q q), that is ( p q), which is in DNF. Conjunctive normal form (CNF) - a formula which is equivalent to a given formula and consists of a product of elementary sums E.g (p q) q ( p q) q is in CNF. 2

3 Sufficient and Necessary Condition A necessary and sufficient condition for an elementary product to be false it contains at least one pair of literals in which one is the negation of the other. A necessary and sufficient condition for an elementary sum to be true it contains at least one pair of liters in which one is the negation of the other. p p F = F p p T = T 3

4 Fallacy or Tautology A given formula is identically false (a contradiction), if every elementary product appearing in its disjunctive normal form is identically false. A given formula in a given formula is identically true (a tautology), if every elementary sum appearing in the formula has at least two literals, of which one is the negation of the other. 4

5 Principal Disjunctive Normal Form Minterms of p and q p q, p q, p q, p q each variable occurs either negated or nonnegated but not both occur together in the conjunction. Principal disjunctive normal form for a given formula, an equivalent formula consisting of disjunctions of minterms only, also called sum of products canonical form. p q = p q = (p q) ( p q) ( p q) What is the principal disjunctive normal form for (p q) ( p r) (q r)? (see example 4) 5

6 Examples of PDNF p q = p q = ( p ( p q)) ((p ( p q)) = ( p) ( p q) ( p p) (p q) =( p ( q q)) (p q) ( p q) =( p q) (p q) ( p q) What is the principal disjunctive normal form for (p q) ( p r) (q r)? (see example 4) (p q)=(p q) (r r)=(p q r) (p q r) ( p r)=( p r q) ( p r q) (q r)=(p q r) ( p q r) Hence, (p q r) (p q r) ( p q r) ( p r q) 6

7 Principal Conjunctive Normal Form Maxterms of p and q p q, p q, p q, p q each variable occurs either negated or nonnegated but not both, appears only once. Principal conjunctive normal form for a given formula, an equivalent formula consisting of disjunctions of maxterms only, also called product-of-sums canonical form. p q = p q is in PCNF. What is the principal conjunctive normal form for [(p q) p q]? (see example 6) 7

8 Examples of PDNF & PCNF p q = (p q) ( p q) in PDNF. p q = [p (q q)] [ q (p p)] = (p q) (p q) ( q p) ( q p) = (p q) (p q) ( p q) in PDNF p q = ( p q) ( q p) in PCNF (p q) (q p) = ( p q) ( q p) = ( p q) ( q p) = = (p q) in PCNF 8

9 Notations of Σ and Π (mutually exclusive) maxterms of p and q p q, p q, p q, p q represented by 00, 10, 01, 11, or Π 0, Π 2, Π 1, Π 3 Minterms of p and q p q, p q, p q, p q represented by 11, 01, 10, 00, or Σ 3, Σ 1, Σ 2, Σ 0 [(p q) p q] =[(p p ) (q p)] q =(q p) q = (q p) q = q p q = p q = Π 1 (p q) (q p) =( p q ) ( q p) = ( p q ) ( q p) =(p q) ( q (p p)) (p (q q)) =(p q) (p q) ( p q) = Σ0,2,3 = Π 1 ( 兩者互補 ) 9

10 First Order Logic Clause form statements whose elementary components are connected by the operator OR ( ) Any statement expressed in the first-order logic can be expressed in clause form. Briefly speaking, each component of a first-order logic expression is expressed by using or or not operators and the whole expression is depicted without implications. 10

11 Prenex Normal Forms (1/2) A formula F is called a Prenex normal form iff F is a first order logic and is in the form of (Q 1 x 1,, Q n x n )(M) where every (Q i x i ), i=1,,n is either ( x i ) or ( x i ) and M is a formula containing no quantifiers. (Q 1 x 1,, Q n x n ) is called the prefix, and M is called the matrix of F. 11

12 Prenex Normal Forms (2/2) Convert a first order logic into prenex normal form 1. Replace and using,, 2. Use double negation and De Morgan s law repeatedly 3. Rename the variables if necessary 4. Use rules of (i) distributes over, distributes over (ii) doesn t distribute over, doesn t distribute over and bring the quantifiers to the left. 12

13 Examples of Prenex Normal Forms (1/2) There are 4 rules which are frequently used: x P(x) x P(x) x P(x) x P(x) 3. (Qx) F(x) G Qx (F(x) G), if G doesn t contain x 4. (Qx) F(x) G Qx (F(x) G), if G doesn t contain x 13

14 Examples of Prenex Normal Forms (2/2) E.g. 1 x P(x) x Q(x) sol: ( x P(x)) x Q(x) x P(x) x Q(x) x ( P(x) Q(x)) E.g. 2 x y ( z (P(x,z) P(y,z)) uq(x,y,u) sol: x y z(p(x,z) P(y,z)) uq(x,y,u) x y ( z P(x,z) P(y,z)) uq(x,y,u) x y z u (( P(x,z) P(y,z)) Q(x,y,u)) x y z u ( P(x,z) P(x,z) Q(x,y,u)) 14

15 Terminology for Proof Axioms statements we assume to be true Proposition, Lemma, Theorem statement that can be shown to be true. Corollary theorem that can be established directly from a proven theorem Conjecture statement that is being proposed to be a true statement, usually based on the basis of some partial evidence, a heuristic argument, or an intuition of an expert. 15

16 Why Proof? Introduction to Proofs. What is a (valid) proof? Why are proofs necessary? 16

17 Introduction to Proof techniques In a proof, one uses axioms/definitions, premises and proven theorems Proof methods: direct, indirect, trivial, contradiction, proof by cases (exhaustive proof), proof of equivalence, existence proofs (constructive or non-constructive), proof by counterexamples, backward/forward reasoning Open Problems famous unsolved problems 17

18 Direct/Indirect Proof A direct proof of a conditional statement p q is constructed when the first step is the assumption that p is true, subsequent steps using rules of inference, with the final step showing q must also be true. Indirect proof if we prove the theorem without starting with the premises and end with the conclusion. E.g. If n is an odd integer, then n 2 is odd. E.g. If n is an integer and 3n+2 is odd, then n is odd. (using indirect proof) 18

19 Proof by Contraposition If (pq) (p+q)/2, then p q Direct proof?? (not trivial) Contrapositive: If p = q, then (pq) = (p+q)/2 It follows by: (pq) = (pp) = (p 2 ) = p (p+p)/2 = (p+q)/2 = p. 19

20 Vacuous and Trivial Proof Vacuous proof in p q, if we know p is false already, the conditional statement must be true. Trivial proof in p q, if we know q is already true. E.g. P(n) is if n > 1, then n 2 > n. Prove P(0) is true. E.g. P(n) is If a and b are positive integers with a b, then a n b n. Prove P(0) is true. 20

21 Proof by cases If n is an integer, then n(n+1)/2 is an integer Case 1: n is even. or n = 2a, for some integer a So n(n+1)/2 = 2a*(n+1)/2 = a*(n+1), which is an integer. Case 2: n is odd. n+1 is even, or n+1 = 2a, for an integer a So n(n+1)/2 = n*2a/2 = n*a, which is an integer. 21

22 Proof by Contradiction 2 is irrational Suppose 2 is rational. Then 2 = p/q, such that p, q have no common factors. Squaring and transposing, p 2 = 2q 2 (even number) So, p is even (if x 2 is even, then x is even) that is, p = 2x for some integer x hence, 4x 2 = 2q 2 or q 2 = 2x 2 So, q is even (if x 2 is even, then x is even) So, p,q are both even they have a common factor of 2. CONTRADICTION. So 2 is NOT rational. Q.E.D. 22

23 Indirect or Contradiction If n is an integer and n 3 +5 is odd, then n is an even Indirect proof (contrapositive) : Let n= k+1, n 3 +5 = 2(4k 3 +6k 2 +3k+3) Proof by Contradiction: Suppose that n 3 +5 is odd and n is odd 5 = (n 3 +5) - n 3 should be even, because of the difference of two odds, but it is an odd. p q: p and q are true, leading to a contradiction. 23

24 Existence Proofs (1/2) E.g.1 There exists (distinct) integers x,y,z satisfying x 2 +y 2 = z 2 Proof: x = 3, y = 4, z = 5. (by constructive existence proof) E.g.2 There is a positive integer that can be written as the sum of cubes of positive integers in two different ways. Proof: 1729 = =

25 Existence Proofs (2/2) There exists irrational b,c, such that b c is rational By nonconstructive proof: Consider 2 2. Two cases are possible: Case 1: 2 2 is rational DONE (b = c = 2). Case 2: 2 2 is irrational Let b = 2 2, c = 2. Then b c = ( 2 2 ) 2 = ( 2) 2* 2 = ( 2) 2 = 2 25

26 The Use of Counterexamples EX1. All prime numbers are odd (false) Proof: 2 is an even number and a prime. EX2. Every prime number can be written as the difference of two squares, i.e. a 2 b 2. Proof: 2 can t be written as a 2 b 2 26

27 Proof by Equivalence n is even iff n 2 is even Proof (by equivalence) Let P be n is even, Q be n 2 is even P and Q are equivalence can be proven by P Q and Q P 27

28 Proofs by Exhaustion Prove that (n+1) 3 3 n if n is a positive integer with n 4 Prove that the only consecutive positive integers not exceeding 100 that are perfect powers are 8 and 9. Show that there are no solutions in integers x and y of x 2 + 3y 2 = 8 Note that: x is perfect power if x = n a, a is a positive integer. 28

29 What is wrong with the proof? If n 2 is positive, then n is positive. Proof: Suppose that n 2 is positive. Because the conditional statement If n is positive, then n 2 is positive is true, hence we can conclude that n is positive. (fallacy of affirming the conclusion) If n is not positive, then n 2 is not positive. Proof: Suppose that n is not positive. Because the conditional statement If n is positive, then n 2 is positive is true, hence we conclude that n 2 is not positive. (fallacy of denying the hypothesis) 29

30 What is wrong with the proof? n is an even integer whenever n 2 is an even integer. Proof: Suppose that n 2 is even. Then n 2 =2k fro some integer k. Let n = 2l for some integer l. This shows that n is even. (fallacy of circular reasoning) Hint: Let n = 2l for some integer l is invalid, since no argument has been given to show that n can be written as 2l for some integer l. 30

31 Conjectures Fermat s Last Theorem x n + y n = z n has no solution in (positive) integers x,y,z with xyz =! 0 whenever n is an integer and is greater than 2. x, y, z, n such that x n + y n = z n? domain of x, y, and z is Z +, domain of n is {x\x > 2, x Z} 31

32 The 3X + 1 Conjecture 3x+1 conjecture Game: Start from a given integer n. If n is even, replace n by n/2. If n is odd, replace n with 3n+1. Keep doing this until you hit 1. e.g. n= Q: Does this game terminate for all n? 32

33 Which Mean? For two positive real numbers x and y, Arithmetic mean Is (x+y)/2 Geometric mean Is Sqrt(xy) (which is less than A-mean) Harmonic mean (G 2 = HA) Is 2/(1/x + 1/y)= 2xy/(x+y) Quadratic mean Is Sqrt((x 2 + y 2 )/2) 33

34 MEMO Read the sections of 1.7 and 1.8. Get familiar with terminology of theorem proving Be familiar with proof methods of contrapositive, counterexample, exhaustive, and existence What is the Fermat s last theorem? HW #13-15,17-19 of 1.7, #13-14,23-24,29-30 of