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## Logic Proofs

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**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**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.**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**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**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**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.**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**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**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.**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!**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**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!**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!**Prove [(t → w) Λ ~ w] → ~ t**[(t → w) Λ ~ w] → ~ t is a Tautology therefore a valid argument!**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**Invalid Arguments**All premises are true but they do not all lead to a valid argument. Example: pq q no conclusion pq ~p no conclusion End for Today**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)**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.**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)**Chain Rule (Law of Syllogism)**[(p → q) Λ(q → r)] → (p → r)**Chain Rule (Law of Syllogism)**[(p → q) Λ(q → r)] → (p → r)**Chain Rule (Law of Syllogism)**[(p → q) Λ(q → r)] → (p → r)**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**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**Chain RuleLaw of Syllogism**p q q r p r [(p → q) Λ(q → r)] → (p → r)**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.**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.**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**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!)**#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.**Negations**• Contrapositive, ~p q ~q ~(~p) ~p q ~q p • Modus Tollens {(~p q) Λ ~q} ~(~p) {(~p q) Λ ~q} p**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)**DeMorgan’s Laws**-Discovered by English mathematician -Tells us how to negate a conjunction and disjunction Complete the following truth table**Since ~(m^s) and ~mV~s are logically equivalent for all**cases, the negation of a conjunction is the disjunction.**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**Laws of Simplification**Mike likes to read and play basketball. We can conclude- Mike likes to read. Mike likes to play basketball.**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**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**Law of Conjunction:**AB is perpendicular to CD AB is the bisector of CD AB is the perpendicular bisector of CD.**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**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.**Other Problems:**Given the statement r, which is a valid conclusion: → 1. r k 2. r k 3. r k 4. r k V ^**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!