Discrete Mathematics

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1 Discrete Mathematics SSDUT, DUT Zhikui Chen, PhD, Professor Office: Office Building 405,Tel: 62274392 Lab: Student Lab

1 1 Discrete Mathematics SSDUT, DUT Zhikui Chen, PhD, Professor Office: Office Building 405,Tel: Lab: Student Lab in Office Building (Big Data Institute) Mobile: QQ:

2 Discrete Mathematics Chapter 1:Propositional Logic

3 2/36 Review Propositional variable Compound formula Tautology Contradiction Tautological implication Substitution rule Replacement rule Basic equivalent formulas (30) Basic tautological implication (16)

4 1.5 Duality principle Definition: P( QR ) P( QR ) P F P T 3/36

5 4/36 Duality principle PQ ( PQ) PQ ( PQ)

6 5/36 Duality principle A B APPBPP (,,,) 12 n (,,,) n APPBPP (,,,) (,,,) 12 n 12 n APPPBPPP (,,,)(,,,) 12 n 12 n ipi ( 1,2,) n A APPBPP (,,,) (,,,) 12 n 12 B n

7 6/36 Duality principle Example: If 若 ( )( ( ) PQPPQ PQ, 试证明 ( PQPPQ )( ( ) PQ Prove:Assume A ( PQ ) ( P( PQ) B PQ try to prove so * A PQ P PQ ( ) ( ( )) * B PQ Because, A B Therefore, * * A B

8 7/36 Duality principle Try to prove: (1)( P Q) ( ) Prove: 证明 :( P Q) ( P Q) 16 PQ T (2)( P Q) ( PQ) F ( P Q) ( P Q) E (( P Q) ( P Q)) ( P Q) E (( P Q) ( P Q)) ( P Q) E, E (( P Q) ( P Q)) (( P Q) ( P Q)) E ( P Q P Q) ( P Q P Q) ( P T ) ( Q T ) E, E T T T E, E E 8

9 8/36 Duality principle Try to prove:(1)( P Q) ( ) Prove: 证明 :( P Q) ( P Q) PQ T (2)( P Q) ( PQ) F (( P Q) ( P Q )) ( P Q) E For 由 ( P Q) ( P Q) (( P Q ) ( P Q)) ( P Q) E, E So 知 ( P Q) ( P Q) 和, ( P Q) ( P Q) 互为对偶式 are reciprocal 由于 T的对偶式是 duality. For F, T is 因此由定理 dual to F, based 知 on theorem 1.5-1, ( P Q) ( P Q) F

10 9/36 Duality principle Theorem 1.5-3: If A B, and A, B are made up of variables P P P B A * * 1, 2,, n and connectives,,,. Prove: A B means is tautological. Based on reverse law, B( P1, P2,, Pn) A( P1, P2,, Pn) is tautological. According to Theorem is tautological. Using the substitution rule, is substituted by P( i 1,2, n), we can get Namely, A( P, P,, P ) B( P, P,, P ) 1 2 n 1 2 B ( P, P,, P ) A ( P, P,, P ) i 1 2 n 1 2 B ( P, P,, P ) A ( P, P,, P ) 1 2 n 1 2 B A n P i n is tautological. n

11 10/ Normal form and judgment Standard form of the formula Normal form Used in the finite steps to determine the formula of permanent true, permanent false, satiable.

12 Disjunctive and Conjunctive normal form Definition: If a propositional formula is the product of some propositional variables and their negations, it's called the basic product;if it is the sum of some variables and their negations, it is called basic sum. Basic product: 基本积 : PQPQPQPQPPQPQ,,,,,,, Basic 基本和 sum: : PPQQPQPQPP,,,,,, A formula composed of the sum of the basic product. If it is equivalent to the given formula A, it is called the disjunctive normal form of A. Disjunctive normal 析取范式 form : : PQPQPQPQPQ,,,,,()() A formula composed of the product of the basic sum. If it is equivalent to the given formula A, it is called the conjunctive normal form of A. 合取范式 :,,,, PQPQPQP,()() 11/36 Conjunctive normal form :

13 Disjunctive and Conjunctive normal form 12/36 Theorem 1.6-1: A basic product is permanent false if and only if it contains two factors likes P, P Prove: (Sufficiency)For P Pis permanent false and QF F, so the basic product that contains and is false. (Necessity) Proof by contradiction: Let the basic product be false but not contains the form of factor P and P, assigning the true value T to the propositional variables in this basic product, and assigning the truth value F to propositional variables with negations, thus the true value of the basic product is T, which contradicts to the hypothesis. P P

14 Disjunctive and Conjunctive normal form 13/36 Theorem1.6-2: A basic sum is permanent true if and only if it contains two factors likes P, P.

15 14/36 Disjunctive and Conjunctive normal form Example:Give the disjunctive normal form of P (Q R) Solution: P (Q R) P ( Q R) ( P Q) ( P R)

16 Disjunctive and Conjunctive normal form 15/36 Example:Give the conjunctive normal form of ( P Q) (P Q) Solution:( P Q) (P Q) (( P Q) (P Q)) ( ( P Q) (P Q)) (( P Q) ( P Q)) ((P Q) (P Q)) ( P Q) (P Q) /*Disjunctive normal formal*/ (( P Q) P) (( P Q) Q) (P Q) ( P Q)

17 Disjunctive and Conjunctive normal form Example:Give the disjunctive normal form of Solution: ( P Q) ( P Q) ( P Q) ( P Q) ( ( P Q)) ( P Q) ( P Q P Q) (( P Q) ( P Q)) F ( P Q) ( P Q) ( P Q) ( P Q) (( P Q) P) (( P Q) Q) P P P Q P Q Q Q F P Q P Q F ( P Q) ( P Q) 16/36

18 17/36 Disjunctive and Conjunctive normal form Example:Give the conjunctive normal form of 解 : 令 A ( P Q) ( P Q), so 那么 Solution: Let A ( ( P Q) ( P Q)) ( ( P Q) ( P Q)) ( ( ( P Q) ( P Q))) (( P Q P Q) (( P Q) ( P Q))) P Q P Q 由于 For A A ( P Q P Q) 所以 A ( P Q) ( P Q) Thus

19 18/36 Principal disjunctive normal form Definition 1.6-4: In the basic product with n variables, if each variable does not coexist with its negation, and one of the two must occur only once, this basic product is called minimum term. Example: The minimum terms for two propositional variables P and Q are: PQPQPQPQ,,, For n propositional variables, the number of minimum terms is 2 n

20 19/36 Principal disjunctive normal form For three variables P, Q, R P Q R P Q R P Q R P Q R P Q R P Q R P Q R P Q R m m m m m m m m

21 20/36 Principal disjunctive normal form Each minimum term has only one truth value assignment to make it T The conjunction of any two minimum terms is false The disjunction for all minimum terms is true

22 21/36 Principal disjunctive normal form Definition 1.6-5:A formula consisting of the sum of the minimum terms, If it is equivalent to a propositional formula A, it is called the principal disjunctive normal form of formula A. The equivalent principal disjunctive normal form can be obtained for any propositional formula (except the permanent false formula), and the form of the principal disjunctive normal form is unique.

23 Principal disjunctive normal form The method to find the principal disjunction normal form: First, it is transformed into its equivalent disjunctive normal form; If the same propositional variable appears more than once in the basic product of the disjunctive normal form, it will be reduced to once only; Remove all basic products that are permanent false in the disjunctive normal form; If a propositional variable is missing from the disjunctive normal form, for example P, ( PPQQ ) is used to add it to the formula. Then the distribution law is employed to expand it and the same basic products are merged. 22/36

24 23/36 Principal disjunctive normal form A P Q R ( P Q ) ( R R ) ( P P ) R ( P Q R ) ( P Q R ) ( P R ) ( P R ) ( P Q R ) ( P Q R ) P R ( Q Q ) ( P R ) ( Q Q ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) ( P Q R ) m 7 m 6 m 5 m 3 m 1 (1, 3, 5, 6, 7 )

25 24/36 Principal disjunctive normal form The relation between Principal disjunctive normal form and truth table The right diagram is the truth table of APQR : Minimum P Q R P Q R terms P Q R P QR PQ R P QR P Q R P QR P Q R P Q R

26 25/36 Principal conjunctive normal form Definition 1.6-6: In the basic sum with n variables, if each variable does not coexist with its negation, and one of the two must occur only once, this basic sum is called maximum term. Example: For two propositional variables P and Q, the maximum terms are: PQPQPQPQ,,, For n variables, the number of maximum terms is 2 n

27 26/36 Principal conjunctive normal form For three variables P, Q, R, P Q R P Q R P Q R P Q R P Q R P Q R P Q R P Q R M M M M M M M M

28 27/36 Principal conjunctive normal form Each maximum term has only one truth value assignment to make it F The disjunction of any two maximum terms is true The conjunction for all maximum terms is false

29 28/36 Principal conjunctive normal form Definition 1.6-7:A formula consisting of the product of the maximum terms, if it is equivalent to a propositional formula A, it is called the principal conjunctive normal form of formula A. The equivalent principal conjunctive normal form can be obtained for any propositional formula (except the permanent true formula), and the form of the principal conjunctive normal form is unique.

30 29/36 Principal conjunctive normal form A P Q R ( P R) ( Q R) (( P R) ( Q Q)) (( Q R) ( P P)) ( P Q R) ( P Q R) ( P Q R) ( P Q R) ( P Q R) ( P Q R) ( P Q R) M M M (0, 2, 4)

31 30/36 Principal conjunctive normal form The relation between principal conjunctive normal form and truth table The right diagram is the truth table of : APQR Considering the minimum terms? Maximum terms P Q R P Q R P Q R P Q R P QR P Q R P QR PQ R P QR P Q R1

32 31/36 Relation between maximum and minimum terms The relations between minimum term and maximum term M i m i M m i i i m M i

33 32/36 From conjunction (disjunction) to principal disjunction (conjunction) They can be transformed into each other For formula A, its principal conjunction is: ( PQRPQR )( ) Give the principal disjunction. Answer: A s principal conjunction is M 1 ΛM 3, so A s principal disjunction is ( 0,2,4,5,6,7 ) Thus, the principal disjunction of A is ( PQR )( PQRPQR )( ) ( PQRPQRPQR )( )( )

34 33/36 Principal disjunction and conjunction For a propositional formula, which is permanent true, all the minimum terms of its propositional variables appear in its principal disjunction, and there is no equivalent principal conjunction; For a propositional formula, which is permanent false, all the maximum terms of its propositional variables appear in its principal conjunction, and there is no equivalent principal disjunction; For a propositional formula, which is permanent satiable, it has the equivalent principal disjunction and the equivalent principal conjunction.

35 34/36 Principal disjunction and conjunction Example: Find the main normal form of the following formula: (P Q) R Answer:(P Q) R ( P Q ) R (P Q) R (P R) (Q R) (P R (Q Q)) (Q R (P P)) (P Q R) (P Q R)) (P Q R) ( P Q R ) M 1,3,5 /* represents for conjunction*/ m 0,2,4,6,7 /*This formula is satiable*/

36 35/36 Principal disjunction and conjunction Example: Find the main normal form of the following formula: (P Q R) ( P Q S) Answer: (P Q R) ( P Q S) (P Q R S) (P Q R S) ( P Q R S) ( P Q R S) m 11, 10, 6, 4 /* represents for disjunction*/ M 0,1,2,3,5,7,8,9,10,12,13,14,15 The principal disjunction and conjunction can be obtained at the same time

37 36/36 Homework P (1)(3) 18(2)(4)

2.2: Logical Equivalence: The Laws of Logic

2.2: Logical Equivalence: The Laws of Logic Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q