CS173 Discrete Mathematical Structures

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CS173Discrete Mathematical Structures

• Cinda Heeren
• Siebel Center rm 2213
• heeren_at_cs.uiuc.edu
• Ofc hr Wednesday, 930a-1130a

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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 groupssocial problem
  solving.

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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/
  

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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 dont drink coffee.
• Some fierce creatures dont drink coffee.

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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.

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Predicates - quantifier negation

• No large birds live on honey.

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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

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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)

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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)

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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)

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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!

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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.

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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!

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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
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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
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CS 173 Proofs - Addition

• I am not a great skater.
• ? I am not a great skater or I am tall.

Inference Rule Addition
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CS 173 Proofs - Simplification

• I am not a great skater and you are sleepy.
• ? you are sleepy.

Inference Rule Simplification
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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
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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
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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.
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CS 173 Proofs - A little proof

• Heres 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
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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)
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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)
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CS 173 Proof Techniques - direct proofs

• A totally different example
• Prove that if n 3 mod 4, then n2 1 mod 4.

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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,
(4k 3)(4k 3)
n2
16k2 24k 9
16k2 24k 8 1
4(4k2 6k 2) 1
4j 1 for some int j
1 mod 4.
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CS 173 Proofs - Fallacies

• Rules of inference, appropriately applied give
  valid arguments.
• Mistakes in applying rules of inference are
  called fallacies.

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CS 173 Proofs - valid arg or fallacy?

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CS 173 Proofs - valid arg or fallacy?