LOGIC
PROPOSITIONAL LOGICl Propositionl declarative statement that is either true or false,but not bothl the following statement...
PROPOSITIONAL LOGICl Notationl atomic propositionsl capital lettersl compound propositionsl atomic propositions with logic...
LOGICAL CONNECTIVESOperator Symbol UsageNegation ¬ notConjunction ∧ andDisjunction ∨ orConditional → if, thenBiconditional...
NEGATIONl turns a false proposition to true and turnsa true proposition to falsel truth tableP ¬ P1 00 1l examplel P: 10 i...
CONJUNCTIONl truth tableP Q P ∧ Q1 1 11 0 00 1 00 0 0
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 pre...
DISJUNCTIONl truth tableP Q P ∨ Q1 1 11 0 10 1 10 0 0
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...
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
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...
CONDITIONAL/IMPLICATIONl examples:l If triangle ABC is isosceles, then the base anglesA and B are equal.l 1+2 = 3 implies ...
BICONDITIONALl logically equivalent to P → Q ∧ Q → Pl truth tableP Q P ↔ Q1 1 11 0 00 1 00 0 1
BICONDITIONALl examplesl A rectangle is a square if and only if its diagonalsare perpendicular.l 5 + 6 = 6 if and only if ...
OTHER CONCEPTSl contrapositivel ¬ Q → ¬ P contrapositive of P → Ql ¬ Q → ¬ P is equivalent to P → Ql inversel ¬ P → ¬ Q is...
OTHER CONCEPTSl types of propositional formsl tautology – a proposition that is always trueunder all possible combinations...
SAMPLE TRUTH TABLESP Q P ∧ Q (P ∧ Q) → P1 1 1 11 0 0 10 1 0 10 0 0 1(P ∧ Q) → P
SAMPLE TRUTH TABLESP ¬ P P ∧ ¬ P1 0 00 1 0P ∧ ¬ P
SAMPLE TRUTH TABLESP Q P ∨ Q (P ∨ Q )→ P1 1 1 11 0 1 10 1 1 00 0 0 1(P ∨ Q) → P
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 ↔ ...
Equivalent Propositions(Logical Equivalence)l When are two propositions equivalent?Suppose P and Q are compound propostion...
Equivalent Propositions(Logical Equivalence)l Suppose P is equivalent to Q. P may be used toreplace Q or vice versa.l The ...
Rules of Replacement1. IdempotenceP ≡ ( P ∨ P ) , P ≡ ( P ∧ P )2. Commutativity( P ∨ Q ) ≡ ( Q ∨ P ), ( P ∧ Q ) ≡ ( Q ∧ P ...
Rules of Replacement5. Distributivity of ∧ over ∨P ∧ ( Q ∨ R ) ≡ ( P ∧ Q ) ∨ ( P ∧ R )6. Distributivity of ∨ over ∧P ∨ ( Q...
Rules of Replacement10. Exportation[ ( P ∧ Q ) ⇒ R ] ≡ [ P ⇒ ( Q ⇒ R ) ]11. Absurdity[ ( P ⇒ Q ) ∧ ( P ⇒ ¬ Q )] ≡ ¬ P12. C...
Rules of Replacement13. IdentitiesP ∨ 1 ≡ 1 P ∧ 1 ≡ PP ∨ 0 ≡ P P ∧ 0 ≡ 0P ∨ ¬P ≡ 1 P ∧ ¬P ≡ 0¬0 ≡ 1 ¬1 ≡ 0
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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

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