(1) Propositional Logic

Transcription

1 King Saud University College of Sciences Department of Mathematics 151 Math Exercises (1) Propositional Logic By: Malek Zein AL-Abidin ه

2 ب) ب) ب) ب) ب) ب) ب) ب) ب) Algebraic Properties of Connectives ) ) Propositions ( بفرض : ) تقارير ب) )1 قاعدتا اإلبدال Rules( ( Commutative : ( أ ) ) )2( قاعدتا التجميع Rules( )Associative : ( أ ) ) : )3( قاعدتا التوزيع Rules( (Distributive ( أ ) ) )4( قاعدتا العنصر المحايد Rules( (Identity : ( أ ) ) ( 5 ) قاعدتا النفي Rules( )Negation : ( أ ) ( 6 ) قاعدة نفي النفي ( Rule ) Double Negation : ) ( 7 ) قاعدتا الجمود ( Rules ) Idempotent : ( أ ) ) ( 8 ) قاعدتا ديمورجان ( Rules ) DeMorgan s : ( أ ) ) ( 9 ) قاعدتا الشمول ( Rules ) Universal : ( أ ) ) )11( قاعدتا اإلمتصاص ( Rules ) Absorption : ( أ ) )11( قاعدتا البرهان البديل ( Rules ) Alternative proof : ( أ ) ب) ) ) )12( قاعدتا الشرط ( Rules ) Conditional : ( أ ) )13( قواعد ثنائي الشرط ( Rules ) Biconditional : ( أ ) ب) ) ج) ) )14( قاعدة المكافئ العكسي ( Contrapositive ) Rule of : )15( قاعدة اإلنطالق والوصول Rule( :) Exportation Importation

3 .)Consequent بينما يسمى التقرير ( النتيجة ( Antecedent في التقرير الشرطي يسمى التقرير )المقدمة يقترن بالتقرير الشرطي تقارير شرطية أخرى هي : العكس ( Converse ) : المعكوس ( Inverse ) : المكافئ العكسي Contrapositive( ) : p q يكون صائبا إذا كان كل منهما صائبا عدا ذلك يكون خاطئا. يكون خاطئا إذا كان كل منهما خاطئا عدا ذلك يكون صوابا. يكون خاطئا صائبا و كان q خاطئا عدا ذلك يكون صائبا. إذا كان p يكون صائبا إذا كان كل منهما صائبا أو إذا كان كل منهما خاطئا عدا ذلك يكون خاطئا. DEINIION 1: A compound proposition that is always true, no matter what the truth values of the propositional variables that occur in it, is called a tautology. A compound proposition that is always false is called a contradiction. A compound proposition that is neither a tautology nor a contradiction is called a contingency. DEINIION 2: he compound propositions p and q are called logically equivalent if p q is a tautology. he notation p q denotes that p and q are logically equivalent.

4 EXAMPLE 2 Show that (p q) and p q are logically equivalent. EXAMPLE 3 Show that p q and p q are logically equivalent. EXAMPLE 4 Show that p (q r) and (p q) (p r) are logically equivalent. his is the distributive law of disjunction over conjunction.

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6 Exercises Q1- Decide whether the following propositions are tautology or a contradiction or a contingency : 1) ( By rules without using the truth tables ) )

7 ( By rules without using the truth tables ) ( Conditional Rule ) ( DeMorgan s Rule) (Commutative and Associative Rules ) ( Negation Rule ) ) (p q) q p q (p q) (p q) q ) [p (p q)] q p q p q p (p q) [p (p q)] q

8 5) (p q) (p q) p q p q p q (p q) (p q) ) (p q) (p q) p q q p q p q (p q) (p q)

9 7) ( By rules without using the truth tables )

10 8) ( By rules without using the truth tables )

11 9) Decide whether the following propositions are tautology or a contradiction? contingincy where 10) Show that the following proposition is a tautology :

12 11) Show that the following proposition is a tautology : 12) Prove the following proposition a tautology, (Don t use the truth table) : Proof:

13 13) Decide whether the following proposition is a tautology (Don t use the truth tables) 14) Show that the following proposition is a tautology : [(p q) (p r) (q r)] r

14 15) Show that the following proposition is a tautology : ( Conditional Rule ) ( Conditional Rule ) ( DeMorgan s Rule) ( DeMorgan s Rule) (Associative Rule) (Distributive Rule) ( Negation Rule ) (Universal Rule) (Identity Rule) ( DeMorgan s & Associative& Negation& Universal Rules) 16) Show that the following proposition is a tautology :

15 17) Show that the following proposition is a contradiction : [ ( p q ) ] [ q r)] 18) Show that the following proposition is a contradiction : [ ( p q ) ( q r ) ] ( p r )

16 Q2 : 1) Show that (p q) and p q are logically equivalent p q q p q (p q) p q (p q) ( p q) ( p) q p q by conditional law by the second De Morgan law by the double negation law 2) Show that (p ( p q)) and p q are logically equivalent by developing a series of logical equivalences. p q p q p q p ( p q) (p ( p q)) p q

17 (p ( p q)) p ( p q) by the second De Morgan law p [ ( p) q] by the first De Morgan law p (p q) by the double negation law ( p p) ( p q) by the second distributive law ( p q) because p p ( p q) by the commutative law for disjunction p q by the identity law for ) Show that - 4) Show that

18 5) Show that - 6) Show that ( Conditional Rule ) (Commutative and Associative Rules ) ( Idempotent Rule ) ( Conditional Rule )

19 7) Show that 8) Show that ( Conditional Rule ) ( DeMorgan s Rule) ( Distributive Rule ) ( Conditional Rule)

20 9) Show that ( Conditional Rule ) ( DeMorgan s Rule) ( Distributive Rule ) ( Conditional Rule) 10) Show that

21 11) Show that 12) Show that ( p q ) ( p r ) and p ( q r ) are logically equivalent?

22 13) Show that ( p r ) ( q r ) and ( p q ) r are logically equivalent? 14) Show that (p q) (p r) and p (q r) are logically equivalent?

23 15) Show that (p r) (q r) and (p q) r are logically equivalent?. 16) Show that p (q r) and q (p r) are logically equivalent?

24 17) Show that p q and (p q) (q p) are logically equivalent ( By the truth table) p q p q q p (p q) (q p) p q 18) Show that p q and p q are logically equivalent?

25 19) Show that 20) Show that

26 21) Show that 22) Show that

27 23) Show that ( Absorption Rule) ( Conditional Rule) ( Distributive Rule) ( Negation Rule) ( Identity Rule ) ( DeMorgan s Rule) ) Show that

28 25) Show that 26) Show that and ( are logically equivalent.

29 27) Decide whether is logically equivalent to or not? (Resolve it by rules)? 28) Decide whether is logically equivalent to or not?

30 29) Show that 30) Show that (Don t use the truth table)

31 31) Show that and are logically equivalent. 32) Show that the contrapositive of is logically equivalent to

32 33) Show that the contrapositive of is logically equivalent to Q3 - State the contrapositive of the following statements : 1) If is an odd number, then is an odd number and also is an odd number. 2) If 3 divides the integers, then 3 divides

33 3) If, then : 4) If the integer is an even, then or or,where 5) If is a prime number where, then is odd. 6) If is integer, then is odd or is even.

34 7) If and are odd integers, then is even. 8) If,then 9) State the converse, the inverse and the contrapositive for these Propositions : A. I will come over whenever there is a football game on. B. I sleep until noon, whenever I stay up late the night before.

35 C. If it is raining, then the home team wins. D. If you solve all exercises then you get a good mark.