lec0126 - CS 173: Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Office Hours: W 9:30-11:30a
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Cs173 - Spring 2004 CS 173 Announcements Homework 1 returned this week. Homework 2 available.  Due 01/29, 8a. Section 10 meets today, 1-2p, in Siebel 2124. Let’s try the clickers… On a typical Friday night I… a. Study b. Drive home c. Go out with friends d. Stay in with friends e. No way I’m telling YOU!!
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CS 173 Proof Techniques - direct proofs Here’s what you know: Ellen is a math major or a CS major. If Ellen does not like discrete math, she is not a CS major. If Ellen likes discrete math, she is smart. Ellen is not a math major. Can you conclude Ellen is smart?  C ((M   C)   ( ¬   ¬ C)   (D   S)   ( ¬ M))   S ?
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Cs173 - Spring 2004 CS 173 Proof Techniques - direct proofs In general, to prove p q, assume p and show that q follows. ((M C) ( ¬ D ¬ C) (D S) ( ¬ M)) S ?
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Cs173 - Spring 2004 CS 173 Proof Techniques - direct proofs 1.  M   C Given 2.  ¬   ¬ C Given 3. D   S Given 4.  ¬ M Given   5. C DS (1,4) 6. D MT (2,5) 7. S MP (3,6) Ellen is smart!
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Cs173 - Spring 2004 CS 173 Proof Techniques - vacuous proofs In general, to prove p   q, assume p and show that q follows.  But p   q is also TRUE if p is FALSE. Suggests proving p   q by proving  ¬ p. Ex.  p: There is good Chinese food in CU. q: I’ll give you each $10. Since p is FALSE, p   q is TRUE  (but we don’t know a thing about q)
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Cs173 - Spring 2004 CS 173 Proof Techniques - trivial proofs In general, to prove p   q, assume p and show that q follows.  But p   q is also TRUE if q is TRUE. Suggests proving p   q by proving q. Ex.  p: there is good Chinese food in CU q: I’m drinking coffee Since q is TRUE, p   q is TRUE  (the truth or falsity of p is irrelevant)
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Cs173 - Spring 2004 CS 173 Proof Techniques - indirect proofs Recall that p   q    ¬   ¬ p (the contrapositive) So, we can prove the implication p   q  by first assuming  ¬ q, and showing  that  ¬ p follows. Example: Prove that if a and b are integers, and a + b ≥ 15, then a ≥ 8 or b ≥  8. (a + b ≥ 15)   (a ≥ 8) v (b ≥ 8)  (Assume  ¬ q) Suppose (a < 8)   (b < 8). (Show  ¬ p) Then (a ≤ 7)   (b ≤ 7), and (a + b) ≤ 14, and (a + b) < 15.
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Cs173 - Spring 2004 CS 173 Proof Techniques - proof by contradiction To prove a proposition p, assume not p and show a contradiction. Suppose the proposition is of the form p   q, and recall that p   q   q v  ¬   ¬ ( ¬  p).  So assuming the opposite is to assume  ¬  p.
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CS 173 Proof Techniques - proof by contradiction Example: Rainy days make gardens grow.
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This note was uploaded on 09/15/2008 for the course CS 173 taught by Professor Fleck@shaffer during the Spring '08 term at University of Illinois at Urbana–Champaign.

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lec0126 - CS 173: Discrete Mathematical Structures Cinda...

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