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Announcements Grading policy No Quiz next week Midterm next week (Th. May 13). The correct answer to the quiz
Now you can debunk these ? (a sample from the web) Humans are mammals. Dogs are mammals. Humans are dogs. Then how am I typing this? 1. Theists (maybe not Jews) define God as all-powerful. 2. Therefore, God can lift any size rock. 3. Therefore, there can be no rock too big for God to lift. 4. Therefore, God cannot create a rock too big for him to lift. 5. Therefore, there is something God cannot do. 6. Therefore, God is not all powerful. 7. Therefore God does not exist (per the definition given by its believers). people make mistakes mistakes are wrong alex is a person alex is wrong
Lecture Methods of Proof
1.5 Some Fallacious Proofs What’s wrong with this?
1.5 Premise 1: If Portland is the capital of Maine, then it is in Maine. Premise 2: Portland is in Maine. Conclusion: Portland is the capital of Maine. Application of the death penalty is killing a human being. Killing a human being is wrong. Therefore, application of the death penalty is wrong.
1.5 In this class we will learn the art of proving theorems. Some names: 1) Theorem, Proposition, Claim, Fact, Result: statement that can be proved. 2) axioms, postulates: the basic assumptions on which the proof us based. 3) lemma: intermediate result to be proved on your way to proof a theorem. 4) corollary: Result that is directly follows from a theorem you just proved. 5) Conjecture: A Result you think is true, but cannot prove. We use rules of inference to prove theorems. By using them wrong, we create fallacious proofs.
1.5 Rules of Inference (modus ponens – law of detachment) always true: it’s a tautology Conclusion: if the premises p and p q are both true, then q can only be true. However, if the premises do not hold, q can still be true or false.
1.5 Examples: it snows today If it snows today we go skiing Therefore: we go skiing If it rains we do not have a barbeque today If we don’t have a barbeque today, we’ll have one tomorrow Therefore: If it rains today, we’ll have a BBQ tomorrow.
1.5 Valid arguments. All inference rules were of the form: premise 1 is true, premise 2 is true, therefore conclusion is true. In general this looks like: For an argument to be true all the premises must be true. Example: if n>1 then n^2 > 1 (True) We cannot conclude (½)^2 > 1 because the premise is not true.
Fallacies (revisited) If you do every problem in this book then you’ll learn discrete math. Joe didnot do every problem in the book, therefore he didnot learn discrete math. correctwrong p = you do all problems in the book. q = you learned discrete math. fallacy of denying hypothesis
Fallacies (revisited) If you do every problem in this book then you’ll learn discrete math. Joe learned discrete math, therefore he did every problem in this book.... correctwrong p = you do all problems in the book. q = do learned discrete math. fallacy of affirming conclusion
1.5 Example: Everyone in this math class has takes a CS course Marla is in this class Therefore: Marla has takes a course in CS D(x) = x has takes a math class C(x) = x has takes a CS class. premises: conclusion: Reasoning: universal instantiation modus ponens
Some more examples Example 10 p.74. All movies produced by John Sayles are wonderful John S. produced a movie about coal-miners Therefore: there a wonderful movie about coalminers. s(x) = x is a movie by John Sayles c(x) = x is a movie about coalminers w(x) = x is a wonderful movie.
Examples If it does not rain or it is not foggy then the sailing race will go on and the lifesaving demonstration will go on. If the sailing is held, then the trophy will be awarded and the trophy was not awarded imply “it rained”. r = it rains f = it is foggy s = sailing race goes on l = lifesaving demonstration goes on. t = trophy awarded. the want the conclusion “r” to be on the right side of the arrow combine premises as much as you can
Examples Example 16. S(x,y) is the statement: x is shorter than y. premise: Therefore: What wrong?
Strategies for proving theorems Direct proof of implication p q Assume p = true and use rules of inference to prove that q is true. Indirect Proof of implication:: Assume q is not true, use rules of inference to prove that p is not true. (NOT q) (NOT p) Proof by contradiction: Assume p is not true and use the rules of inference to prove a contradiction. (NOT p) False
Direct/Indirect Proofs Proof the following theorem: If n is an odd integer, then n^2 is an odd integer. assume p (n is an odd integer). n = 2k+1 for some integer k, Then: n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2 ( 2k^2 + 2k) + 1 = 2m+1, m integer Proof that if 3n+2 is odd, then n = odd. Assume (NOT q) : n = even. Then n = 2k, 3n+2 = 6k + 2 = 2(3k+1) = 2m. Thus, 3n+2 is even. We have proved (NOT q) (NOT p) which is equivalent to p q
Contradiction Prove: p: At least 4 of any 22 days must fall on the same day of the week (the pigeonhole principle !). Assume (NOT p). Then at most 3 days of any 22 are the same day of the week. This implies that we could only have chosen 3x7=21 days, which is a contradiction with the fact that we had chosen 22 days to begin with. Thus (NOT p) = False p = True.