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# Propositional logic

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Discrete Structures

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### Propositional logic

1. 1. PREPOSITIONal LOGIC
2. 2. A statement is a declaratory sentence which is true orfalse but not both. In other words , a statement is adeclarative sentence which has a definate truth table.
3. 3. Logical connectives or sentenceconnectives These are the words or symbols used to combine two sentence to form a compound statement. logic Name rank ~ Negation 1 ^ Conjunction 2 V Disjunction 3 => Conditional 4  Biconditional 5
4. 4. A B ^ V ~A =>  NOR NAND XOR EX- NORT T T T F T T F F F TT F F T F F F F T T FF T F T T T F F T T FF F F F T T T T T F T
5. 5. TAUTOLOGYi. A TAUTOLOGY IS A PREPOSITION WHICH IS TRUE FOR ALL TRUTH VALUES OF ITS SUB- PREPOSITIONS OR COMPONENTS.ii. A TAUTOLOGY IS ALSO CALLED LOGICALLY VALID OR LOGICALLY TRUE.iii. ALL ENTRIES IN THE COLUMN OF TAUTOLOGY ARE TRUE.
6. 6. For example: p^q=>qP q p^q q p^q=> qT T T T TT F F F TF T F T TF F F F T
7. 7. Contradiction CONTRADICTION IS A PREPOSITION WHICH IS ALWAYS FALSE FOR ALL TRUTH VALUES OF ITS SUB-PREPOSITIONS OR COMPONENTS. A CONTRADICTION IS ALSO CALLED LOGICALLY INVALID OR LOGICALLY FALSE ALL ENTRIES IN THE COLUMN OF CONTRADICTION ARE FALSE.
8. 8. FOR EXAMPLE (P v Q)^(~P)^(~Q)P Q PVQ ~P ~Q (P v Q)^(~P)^(~Q)T T T F F FT F T F T FF T T T F FF F F T T F
9. 9. Contingency It is a preposition which is either true orfalse depending on the truth value of its components or preposition..
10. 10. FOR EXAMPLE ~p ^ ~qp q ~p ~q ~p ^ ~qT T F F FT F F T FF T T F FF F T T T
11. 11. Logical equivalenceTwo statements are called logically equivalent if the truthvalues of both the statements are always identical.. For example: If we take two statements p=>q and ~q =>~p , then theretruth table values must be equal to satisfy the condition oflogical equivalence..
12. 12. p q ~p ~q p=>q ~q=>~p T T F F T T T F F T F F F T T F T T F F T T T T SINCE,THE TRUTH TABLE VALUES OF BOTH STATEMENTS IS SAME. THUS, THE TWOSTATEMENTS ARE LOGICALLY EQUIVALENT..
13. 13. LOGICAL IMPLICATIONS DIRECT IMPLICATION (p=>q) CONVERSE IMPLICATION (q=>p) INVERSE OR OPPOSITE IMPLICATION (~p=>~q) CONTRAPOSITIVE IMPLICATION (~q=>~p)
14. 14. Algebra of preposition1) Commutative law2) Associative law3) Distributive law4) De Morgan’s law5) Idempotent law6) Identity law
15. 15. Idempotent law 1. pVpp 2. p^ppp p pvp p v pp p^p p^ ppT T T T T TF F F F F F
16. 16. Commutative law • pvq=qvp • p^q=q^pp q pvq qvp p^q q^pT T T T T TT F T T F FF T T T F FF F F F F F
17. 17. Associative law• (p v q) v r  p v (q v r)• (p ^ q) ^ r  p ^ (q ^ r) p q r pvq ( p v q) v r qVr p v (q v r) T T T T T T T T T F T T T T T F T T T T T T F F T T F T F T T T T T T F T F T T T T F F T F T T T F F F F F F F
18. 18. Distributive law• p ^ (q v r)  (p ^ q) v (p ^ r)• p ^ (q v r)  (p ^ q) v (p ^ r) p q r qvr p^(q v r) p^q p^r (p^q)v(p^r) T T T T T T T T T T F T T T F T T F T T T F T T T F F F F F F F F T T T F F F F F T F T F F F F F F T T F F F F F F F F F F F F
19. 19. De Morgan’s law• ~(p v q)  ~p ^ ~q• ~(p ^ q)  ~p v ~q p q (p v q) ~(p v q) ~p ~q ~p ^ ~q T T T F F F F T F T F F T F F T T F T F F F F F T T T T
20. 20. Identity law1) p ^ T  p 2) T ^ p  p3) p v F  p 4) F v p  pP T P^T P F P v FT T T T F TF T F F F F
21. 21. TRANSITIVE RULE pq qr -------------- prRule of detachment P Pq ---------- q
22. 22. EXAMPLE TEST THE VALIDITY OF THE FOLLOWING ARGUMENT…. IF A MAN IS A BACHELOR,HE IS WORRIED(A PREMISE) IF A MAN IS WORRIED,HE DIES YOUNG(A PREMISE)----------------------------------------------------------------------------------------------------- BACHELORS DIE YOUNG(CONCLUSION) P: A man is a bachelor Q:he is worried R: he dies young
23. 23. The given argument in symbolic form can bewritten as: pq (a premise) qr (a premise) -------------------- pr (conclusion) The given argument is true by law ofsyllogism(law of transitive)…
24. 24. p q r pq qr pr pq ^ qr (pq) ^ (qr) => prT T T T T T T TT T F T F F F TT F T F T T F TT F F F T F F TF T T T T T T TF T F T F T F TF F T T T T T TF F F T T T T T
25. 25. PRESENTATION BY : ASHWINI VIPAT