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Unformatted text preview: Administration Piazza class forms I I have some and will hand out with the following priority. 1. Richard Dore’s Tuesday section. 2. Attended section and account forms ran out. 3. Everyone else. I I am having more printed..and will advise on piazza about.. 1. how to get a form and 2. what to do with electronic sub if you don’t. CS70: Lecture 4. Outline. 1. More practice with proofs. 2. Induction. 3. Simple Proof: 5 year old Gauss. 4. A horse with no name.. 5. Two coloring map 6. Strengthening induction. 7. Try this at home. Contradiction Theorem: √ 2 is irrational. Contradiction Theorem: √ 2 is irrational. Assume it is rational: Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 a 2 is even = ⇒ a is even. Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 a 2 is even = ⇒ a is even. a = 2 k for some integer k Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 = 4 k 2 a 2 is even = ⇒ a is even. a = 2 k for some integer k Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 = 4 k 2 a 2 is even = ⇒ a is even. a = 2 k for some integer k b 2 = 2 k 2 Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 = 4 k 2 a 2 is even = ⇒ a is even. a = 2 k for some integer k b 2 = 2 k 2 b 2 is even = ⇒ b is even. Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 = 4 k 2 a 2 is even = ⇒ a is even. a = 2 k for some integer k b 2 = 2 k 2 b 2 is even = ⇒ b is even. a and b have a common factor. Contradiction. Contradiction Theorem: √ 2 is irrational. Assume it is rational: √ 2 = a / b . Reduced form: a and b have no common factors. √ 2 b = a 2 b 2 = a 2 = 4 k 2 a 2 is even = ⇒ a is even....
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 Fall '11
 Rau
 Mathematical Induction, Natural number, Mathematical logic, base case, Mathematical proof

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