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CS173 Discrete Mathematical Structures

CS173 Discrete Mathematical Structures. Cinda Heeren Siebel Center rm 2213 heeren@cs.uiuc.edu Ofc hr: Wednesday, 9:30a-11:30a. Announcements. How did hwk #1 go? Anyone want to share? Hwk #2 available. Due Sun, 1/29, 8a. Sections start this week! See course website for locations.

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CS173 Discrete Mathematical Structures

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  1. CS173Discrete Mathematical Structures Cinda Heeren Siebel Center rm 2213 heeren@cs.uiuc.edu Ofc hr: Wednesday, 9:30a-11:30a

  2. Announcements • How did hwk #1 go? Anyone want to share? • Hwk #2 available. Due Sun, 1/29, 8a. • Sections start this week! See course website for locations. • Look at hwk #2 before you go to section. • Be prepared to work in groups…social problem solving.

  3. Miscellaneous • Text: Rosen • RF devices: (in bookstores) • Automated attendance • Class participation (for fun and feedback) • Class keys: • Section M: J16787I481 • Register for course at http://www.einstruction.com • Web: http://www.cs.uiuc.edu/class/cs173 • IRC chat room: http://www.quickfire.org/cs173 • Class wiki: https://www-s.cs.uiuc.edu/wiki/cs173/

  4. x (L(x)  F(x)) Universe of discourse is all creatures. x (L(x)  C(x)) x (F(x)  C(x)) Predicates - more examples L(x) = “x is a lion.” F(x) = “x is fierce.” C(x) = “x drinks coffee.” All lions are fierce. Some lions don’t drink coffee. Some fierce creatures don’t drink coffee.

  5. Predicates - quantifier negation So, x P(x) is the same as x P(x). So, x P(x) is the same as x P(x). General rule: to negate a quantifier, move negation to the right, changing quantifiers as you go.

  6. What’s wrong with this proof? Predicates - quantifier negation No large birds live on honey.

  7. Reminder: in a proposition, all variables must be bound. Predicates - free and bound variables A variable is bound if it is known or quantified. Otherwise, it is free. Examples: P(x) x is free P(5) x is bound to 5 x P(x) x is bound by quantifier

  8. True proposition • False proposition • Not a proposition • No clue c) b) a) b) Predicates - multiple quantifiers To bind many variables, use many quantifiers! Example: P(x,y) = “x > y” • x P(x,y) • xy P(x,y) • xy P(x,y) • x P(x,3)

  9. P(x,y) true for all x, y pairs. For every value of x we can find a (possibly different) y so that P(x,y) is true. P(x,y) true for at least one x, y pair. There is at least one x for which P(x,y) is always true. quantification order is not commutative. Suppose P(x,y) = “x’s favorite class is y.” Predicates - the meaning of multiple quantifiers • xy P(x,y) • xy P(x,y) • xy P(x,y) • xy P(x,y)

  10. False True? True False Predicates - the meaning of multiple quantifiers N(x,y) = “x is sitting by y” • xy N(x,y) • xy N(x,y) • xy N(x,y) • xy N(x,y)

  11. How do we know it? Proofs - how do you know? The following statements are true: If I am Mila, then I am a great swimmer. I am Mila. What do we know to be true? I am a great swimmer!

  12. Axiom, postulates, hypotheses and previously proven theorems. Rules of inference Proof Proofs - how do you know? A theorem is a statement that can be shown to be true. A proof is the means of doing so.

  13. What rule of inference can we use to justify it? Proofs - how do you know? The following statements are true: If I am Mila, then I am a great swimmer. I am Mila. What do we know to be true? I am a great swimmer!

  14. p p  q Tautology: (p  (p  q))  q  q CS 173 Proofs - Modus Ponens I am Mila. If I am Mila, then I am a great swimmer.  I am a great swimmer! Inference Rule: Modus Ponens

  15. q p  q Tautology: (q  (p  q))  p  p CS 173 Proofs - Modus Tollens I am not a great skater. If I am Erik, then I am a great skater.  I am not Erik! Inference Rule: Modus Tollens

  16. p Tautology: p  (p  q)  p  q CS 173 Proofs - Addition I am not a great skater.  I am not a great skater or I am tall. Inference Rule: Addition

  17. p  q Tautology: (p  q) p  p CS 173 Proofs - Simplification I am not a great skater and you are sleepy.  you are sleepy. Inference Rule: Simplification

  18. p  q q Tautology: ((p  q)  q)  p  p CS 173 Proofs - Disjunctive Syllogism I am a great eater or I am a great skater. I am not a great skater.  I am a great eater! Inference Rule: Disjunctive Syllogism

  19. p  q q  r Tautology: ((p  q)  (q  r))  (p  r)  p  r CS 173 Proofs - Hypothetical Syllogism If you are an athlete, you are always hungry. If you are always hungry, you have a snickers in your backpack.  If you are an athlete, you have a snickers in your backpack. Inference Rule: Hypothetical Syllogism

  20. Addition Modus Tollens CS 173 Proofs - A little quiz… Amy is a computer science major.  Amy is a math major or a computer science major. If Ernie is a math major then Ernie is geeky. Ernie is not geeky!  Ernie is not a math major.

  21. CS 173 Proofs - A little proof… 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? M  C D  C D  S M

  22. Ellen is smart! CS 173 Proofs - A little proof… 1. M  C Given 2. D  C Given 3. D  S Given 4. M Given 5. C DS (1,4) 6. D MT (2,5) 7. S MP (3,6)

  23. Ellen is smart! CS 173 Proofs - A little proof… 1. M  C Given 2. D  C Given 3. D  S Given 4. M Given 5. C Disjunctive Syllogism (1,4) 6. C  D Contrapositive of 2 7. C  S Hypothetical Syllogism (6,3) 8. S Modus Ponens (5,7)

  24. HUH? 7 = 3 mod 4 37 = 1 mod 4 94 = 2 mod 4 16 = 0 mod 4 7 = 111 mod 4 37 = 61 mod 4 94 = 6 mod 4 16 = 1024 mod 4 CS 173 Proof Techniques - direct proofs A totally different example: Prove that if n = 3 mod 4, then n2 = 1 mod 4.

  25. CS 173 Proof Techniques - direct proofs A totally different example: Prove that if n = 3 mod 4, then n2 = 1 mod 4. If n = 3 mod 4, then n = 4k + 3 for some int k. But then, n2 = (4k + 3)(4k + 3) = 16k2 + 24k + 9 = 16k2 + 24k + 8 + 1 = 4(4k2 + 6k + 2) + 1 = 4j + 1 for some int j = 1 mod 4.

  26. CS 173 Proofs - Fallacies Rules of inference, appropriately applied give valid arguments. Mistakes in applying rules of inference are called fallacies.

  27. If I am Bonnie Blair, then I skate fast I skate fast! If you don’t give me $10, I bite your ear. I bite your ear!  I am Bonnie Blair  You didn’t give me $10. I’m Eric Heiden I’m just mean. Affirming the conclusion. ((p  q)  q)  p Not a tautology. CS 173 Proofs - valid arg or fallacy?

  28. If it rains then it is cloudy. It does not rain. If it is a car, then it has 4 wheels. It is not a car.  It is not cloudy  It doesn’t have 4 wheels. February! ATV Denying the hypothesis. ((p  q)  p)  q Not a tautology. CS 173 Proofs - valid arg or fallacy?

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