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Logic Proofs

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  1. Logic Proofs

  2. Monday 9/29/08 • Take out HW – “Drawing Conclusions”; Pick up handouts! • Quiz handed back Wednesday • Logic Proofs • - Law of Detachment • HW – 1. Essay Due Thursday! • 2. Law of Detachment

  3. Definitions: Logically equivalent: when two statements always have the same truth value. Premise: the statement that is given and excepted to be true. Conclusion: the statement that has come from the premises.

  4. Law of Detachment(a law of inference) • The Law of Detachment states that when two given premises are true, one a conditional and the other the hypothesis of that conditional it then follows that the conclusion of the conditional is true. • If you are given a conditional and the hypothesis the conclusion is true. • p q and p therefore q End for today

  5. Wednesday 10/1/08 • Take out HW – “Law of Detachment”; Pick up handout! • Place #1 – 4 answers (truth tables) on board! HW pass • 2. Discuss Quiz • 3. Logic Proofs • - The Law of Contrapositive • - Formal Proof • - Modus Tollens • HW – 1. Essay Due Thursday! • 2. Law of Contrapositive / Modus Tollens

  6. Law of Contrapositive • The Law of Contrapositive states that when a conditional premise is true, it follows that the contrapositive of the premise is also true. • If the conditional is true the contrapositive is also true. • p  q then ~q  ~p

  7. Formal Proof: given premises that are true use laws of reasoning to reach a conclusion. Two Column Proof: Column 1: Statements Column 2: Reasons Each statement must have a reason and they are number in sequence.

  8. Example: Given: If Joanna saves enough money, then she can buy a bike. Joanna can not buy a bike. Prove: Joanna did not have enough money. Let m = “ Joanna saves enough money” Let b = “ Joanna can buy a bike” Given: m  b, ~b Prove: ~m

  9. Statement m  b ~b Reason Given Given 3. Law of Contrapositive (step 1) 3. ~b  ~m 4. ~m 4. Law of Detachment (step 2 & 3) End for today

  10. Quiz Review 1. Rewrite the following as an equivalent DISJUNCTION! a. 2x ≠ 4 a. (2x < 4) ∨ (2x > 4) b. (b - 2 < 2) ∨ (b - 2 > 2) b. b - 2 ≠ 2 • Write the negation of each in simplest terms! • A. 10 + x < 5 • B. I never fail quizzes. • C. Math is logical. • D. It is not true that I am a good baseball player. A. 10 + x ≥ 5 B. I sometimes fail quizzes C. Math is not Logical. D. I am a good baseball player.

  11. Quiz Review • Write the following in symbolic form using the letters given. In each case the letter is the positive value • of the statement! a. The absolute value of x is equal to 2 if and only if x =2 or x = -2. (p: absolute value of x is 2; q: x = 2; r: x = -2) b. If I’m late, then I’ll get into trouble, and if I’m not late, then I won’t get into trouble.. (l,t) Answers placed on board!

  12. Thursday 10/2/08 • Take out HW – “Law of Contrapositive”; Place essay in box! • Complete Pre-game Warm-up! See Quiz Review • 3. Logic Proofs • - Modus Tollens • - Chain Rule (Law of Syllogism) • HW – 1. Law of Contrapositive / Modus Tollens #21-30 • 2. Complete Chain Rule Worksheet

  13. Law of Modus Tollens Given: If the tickets are sold out (t), then we’ll wait for the next show. (w) We do not wait for the next show. Prove: The tickets were not sold out. Write the problem in symbolic notation!

  14. Law of Modus Tollens Given:t →w t → w ~w ~ w Prove:~ t ~ t or [(t → w) Λ ~ w] → ~ t Set up a truth table to prove!

  15. Prove [(t → w) Λ ~ w] → ~ t

  16. Prove [(t → w) Λ ~ w] → ~ t [(t → w) Λ ~ w] → ~ t is a Tautology therefore a valid argument!

  17. Law of Modus Tollens • The law of Modus Tollens states when 2 given premises are true, one a conditional and the other the negation of the conclusion of that conditional, it then follows that the negation of the hypothesis of the conditional is true. • Given a conditional and the negation of the conclusion then the negation of the hypothesis is true. (combining Law of Contrapositive and Law of Detachment) • p  q, ~q therefore ~p

  18. Invalid Arguments All premises are true but they do not all lead to a valid argument. Example: pq q no conclusion pq ~p no conclusion End for Today

  19. Friday 10/3/08 • Take out HW – “Law of Contrapositive, Chain Rule worksheet” • Complete Quiz Retake! See Quiz Reviewhandout • 3. Logic Proofs • - Discuss Valid arguments • - Chain Rule (Law of Syllogism); Applications of chain rule • - Law of Disjunctive Inference • HW – 1. Law of Disjunctive Inference worksheet • 2. Application of Chain Rule #2-40 (evens)

  20. Quiz Review 1. Rewrite the following as an equivalent DISJUNCTION! -x + 5 ≠ -3 • Write the negation of each in simplest terms! • A. x ≤ -5 • B. Monkeys do not live in trees. • C. Math is never logical.

  21. Quiz Review • Write the following in symbolic form using the letters given. In each case the letter is the positive value • of the statement! If I don’t go to practice and train hard, then I will not be prepared for the game next week and our team will not win (p,h,g,w)

  22. Chain Rule (Law of Syllogism) [(p → q) Λ(q → r)] → (p → r)

  23. Chain Rule (Law of Syllogism) [(p → q) Λ(q → r)] → (p → r)

  24. Chain Rule (Law of Syllogism) [(p → q) Λ(q → r)] → (p → r)

  25. Valid Arguments An argument is valid if the implication (P1 P2 P3 P4 …. Pn) C.  Λ Λ Λ Λ Λ Premises Premises Is a valid argument? [(p → q) Λ ~ q] → ~ p Law of Modus Tollens

  26. Chain RuleLaw of Syllogism • Chain rule states that when 2 given premises are true conditional such that the consequent (conclusion) of the 1st is the antecedent (hypothesis) of the 2nd it follows that a conditional formed using the antecedent of the 1st and the consequent of the 2nd is true. • You can combine conditionals if the conclusion of one is the hypothesis of another. • p  q and q r then p  r

  27. Chain RuleLaw of Syllogism p q q  r p  r [(p → q) Λ(q → r)] → (p → r)

  28. Chain RuleExample p : You study q : You pass r : You get a surprise P1: p  q If you study, then you will pass. P2: q  r If you pass, then you will get a surprise.

  29. Chain RuleExample p : You study q : You pass r : You get a surprise P1: p  q If you study, then you will pass. P2: q  r If you pass, then you will get a surprise. p  r C: If you study, then you will get a surprise.

  30. Law of Disjunction • The previous laws involved conditionals this one does not. • Law of Disjunctive Inference states when 2 given premises are true, on a disjunction and the other the negation of one of the disjuncts it then follows the other disjunct is true. p ν q or p ν q ~q ~p  p  q End for Today

  31. Monday 10/6/08 • Take out HW – “Law of Disjuctive Inference, Chain Rule worksheet & Chain Rule Applications” – Place proofs on the white board! • Another Proof - #10 Disjunctive Inference • 3. Logic Proofs • - Double Negation • - DeMorgan’s Law • 4. Begin HW; Pass back Quizzes • HW – 1. pg 86-87 (evens) • 2. Parent Signatures (3/2- by tomorrow!)

  32. #10 Hw Disjunctive Inference Givens: Don is first or Nancy is Second. If Nancy is second, then Chris is third. If Chris is third, then Pattie is fourth. Pattie is not fouth. Prove: Pattie is not fourth.

  33. Negations • Contrapositive, ~p q  ~q ~(~p) ~p q  ~q  p • Modus Tollens {(~p q) Λ ~q}  ~(~p) {(~p q) Λ ~q}  p

  34. Law of Double Negation • Law of double negation states ~(~p) and p are logically equivalent. • (you do not need to apply this law, continue as we have been)

  35. DeMorgan’s Laws -Discovered by English mathematician -Tells us how to negate a conjunction and disjunction Complete the following truth table

  36. Since ~(m^s) and ~mV~s are logically equivalent for all cases, the negation of a conjunction is the disjunction.

  37. DeMorgan’s Law states • The negation of a conjunction of 2 statements is logically equivalent to the disjunction of the negation of each of the two statements. • The negation of a disjunction of 2 statements is logically equivalent to the conjunction of the negation of each of the 2 statements. • ~(p ^ q)  (~p V ~q) • ~(pV q )   (~p ^~q) End for today

  38. Laws of Simplification Mike likes to read and play basketball. We can conclude- Mike likes to read. Mike likes to play basketball.

  39. Laws of Simplification • The law of simplification states that when a single conjunctive premise is true, it follows that each of the individual conjucts must be true. • p ^ q therefore p is true • p ^ q therefore q is true

  40. Law of Conjunction: The law of conjunction states that when 2 given premises are true, it follows that the conjunction of these is true. p q p ^ q

  41. Law of Conjunction: AB is perpendicular to CD AB is the bisector of CD AB is the perpendicular bisector of CD.

  42. Law of Disjunctive Addition • Law of disjunctive addition states that when a single premise is true, it follows that any disjunction that has this premise as a disjunct is also true. p p V q

  43. Law of Disjunctive Addition We solve an equation and see that x = 5. We may also include that the statement x = 5 or x > 5 is True. We may also include that the statement x = 5 or Ian has 3 eyes is True.

  44. Other Problems: Given the statement r, which is a valid conclusion: → 1. r k 2. r k 3. r k 4. r k V ^

  45. Wednesday 10/8/08 • Take out HW – “Law of Simplification,…” – Place proofs on the white board! • 2. Logic Proofs – Practice!!! • HW – Logic Proof Quiz Friday!