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# Logic parti

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### Logic parti

1. 1. LOGIC
2. 2. PROPOSITIONAL LOGICl Propositionl declarative statement that is either true or false,but not bothl the following statements are propositions:l The square root of 2 is irrational.l In the year 2010, more Filipinos will go to Canada.l -5 < 75l the following statements are NOT propositions:l What did you say?l This sentence is false.l x = 6
3. 3. PROPOSITIONAL LOGICl Notationl atomic propositionsl capital lettersl compound propositionsl atomic propositions with logical connectivesl defined via truth tablesl truth values of propositionsl 1 or T (true)l 0 or F (false)
4. 4. LOGICAL CONNECTIVESOperator Symbol UsageNegation ¬ notConjunction ∧ andDisjunction ∨ orConditional → if, thenBiconditional ↔ iff
5. 5. NEGATIONl turns a false proposition to true and turnsa true proposition to falsel truth tableP ¬ P1 00 1l examplel P: 10 is divisible by 2.l ¬ P: 10 is not divisible by 2.
6. 6. CONJUNCTIONl truth tableP Q P ∧ Q1 1 11 0 00 1 00 0 0
7. 7. CONJUNCTIONl examplesl 6 < 7 and 7 < 8l 2*4 = 16 and a quart is larger than a liter.l P: Barrack Obama is the American president.Q: Benigno Aquino III is the Filipino president.R: Corazon Aquino was an American president.P ∧ Q P ∧ R R ∧ Q
8. 8. DISJUNCTIONl truth tableP Q P ∨ Q1 1 11 0 10 1 10 0 0
9. 9. DISJUNCTIONl examplesl 6 < 7 or Venus is smaller than earth.l 2*4 = 16 or a quart is larger than a liter.l P: Slater Young is a millionaire.Q: Lucio Tan is a billionaireR: Steve Jobs was a billionaire.P ∨ Q P ∨ R R ∨ Q
10. 10. CONDITIONAL/IMPLICATIONl P is the hypothesis or premisel Q is the conclusionl truth tableP Q P → Q1 1 11 0 00 1 10 0 1
11. 11. CONDITIONAL/IMPLICATIONl other ways to express P → Q:l If P then Ql P only if Ql P is sufficient for Ql Q if Pl Q whenever Pl Q is necessary for P
12. 12. CONDITIONAL/IMPLICATIONl examples:l If triangle ABC is isosceles, then the base anglesA and B are equal.l 1+2 = 3 implies that 1 < 0.l If the sun shines tomorrow, I will play basketball.l If you get 100 in the final exam, then you willpass the course.l If 0 = 1, then 3 = 9.
13. 13. BICONDITIONALl logically equivalent to P → Q ∧ Q → Pl truth tableP Q P ↔ Q1 1 11 0 00 1 00 0 1
14. 14. BICONDITIONALl examplesl A rectangle is a square if and only if its diagonalsare perpendicular.l 5 + 6 = 6 if and only if 7 + 1 = 10.
15. 15. OTHER CONCEPTSl contrapositivel ¬ Q → ¬ P contrapositive of P → Ql ¬ Q → ¬ P is equivalent to P → Ql inversel ¬ P → ¬ Q is the inverse of P → Ql P → Q is not equivalent to its inversel conversel Q → P is the converse of P → Q
16. 16. OTHER CONCEPTSl types of propositional formsl tautology – a proposition that is always trueunder all possible combinations of truth valuesfor all component propositionsl contradiction – a proposition that is always falseunder all possible combinations of truth valuesfor all component propositionsl contingency – a proposition that is neither atautology nor a contradiction
17. 17. SAMPLE TRUTH TABLESP Q P ∧ Q (P ∧ Q) → P1 1 1 11 0 0 10 1 0 10 0 0 1(P ∧ Q) → P
18. 18. SAMPLE TRUTH TABLESP ¬ P P ∧ ¬ P1 0 00 1 0P ∧ ¬ P
19. 19. SAMPLE TRUTH TABLESP Q P ∨ Q (P ∨ Q )→ P1 1 1 11 0 1 10 1 1 00 0 0 1(P ∨ Q) → P
20. 20. SAMPLE TRUTH TABLESP Q P ↔ Q P ∧ Q ¬ P ∧¬ Q (P ∧ Q) ∨(¬ P ∧¬ Q)1 1 1 1 0 11 0 0 0 0 00 1 0 0 0 00 0 1 0 1 1Show that (P ↔ Q) ↔(( P ∧ Q) ∨ (¬ P ∧¬ Q))
21. 21. Equivalent Propositions(Logical Equivalence)l When are two propositions equivalent?Suppose P and Q are compound propostions, Pand Q are equivalent if the truth value of P isalways equal to the truth value of Q for all thepermutation of truth values to the componentpropositions
22. 22. Equivalent Propositions(Logical Equivalence)l Suppose P is equivalent to Q. P may be used toreplace Q or vice versa.l The Rules of Replacement are equivalentpropositions(Logically equivalent propositions)l The Rules of Replacement are used to simplify aproposition (Deriving a proposition equivalent toa given proposition)
23. 23. Rules of Replacement1. IdempotenceP ≡ ( P ∨ P ) , P ≡ ( P ∧ P )2. Commutativity( P ∨ Q ) ≡ ( Q ∨ P ), ( P ∧ Q ) ≡ ( Q ∧ P )3. Associativity,( P ∨ Q ) ∨ R ≡ P ∨ ( Q ∨ R ),( P ∧ Q ) ∧ R ≡ P ∧ ( Q ∧ R )4. De Morgan’s Laws¬ ( P ∨ Q ) ≡ ¬P ∧ ¬Q,¬ ( P ∧ Q ) ≡ ¬P ∨ ¬Q
24. 24. Rules of Replacement5. Distributivity of ∧ over ∨P ∧ ( Q ∨ R ) ≡ ( P ∧ Q ) ∨ ( P ∧ R )6. Distributivity of ∨ over ∧P ∨ ( Q ∧ R ) ≡ ( P ∨ Q ) ∧ ( P ∨ R )7. Double Negation¬ (¬ P) ≡ P8. Material Implication( P ⇒ Q ) ≡ (¬ P ∨ Q )9. Material Equivalence( P ⇔ Q ) ≡ ( P ⇒ Q ) ∧ ( Q⇒P )
25. 25. Rules of Replacement10. Exportation[ ( P ∧ Q ) ⇒ R ] ≡ [ P ⇒ ( Q ⇒ R ) ]11. Absurdity[ ( P ⇒ Q ) ∧ ( P ⇒ ¬ Q )] ≡ ¬ P12. Contrapositive( P ⇒ Q ) ≡ (¬ Q ⇒ ¬P )
26. 26. Rules of Replacement13. IdentitiesP ∨ 1 ≡ 1 P ∧ 1 ≡ PP ∨ 0 ≡ P P ∧ 0 ≡ 0P ∨ ¬P ≡ 1 P ∧ ¬P ≡ 0¬0 ≡ 1 ¬1 ≡ 0