Lesson 4: Proofs

1
Lesson 4 Proofs

• Objectives
• Explain the Rules of Inference (page 58)
• Develop Valid Arguments Based on Rules of
  Inference
  
• Evaluate the Validity of Arguments
• Use the Rules of Inference for Quantified
  Statements in proofs
  
• Perform (and identify) direct, indirect, vacuous,
  and trivial proofs
  

• Outline
• Terminology
• Rules of Inference
• Proving Valid Arguments
• Proof Techniques
• Reading Section 1.5
• HW due Feb 1
• 1.5 2, 5, 7, 20, 22, 36
• 1.6 2, 7, 9, 13, 21, 24ac

2
Perspective

• Build a man a fire, and he stays warm for an
  hour
  
• Set a man on fire, and he stays warm the rest of
  his life.

3
Definitions

• Theorem A statement of fact proved to be true
• Proof A sequence of logical statements that
  form a valid argument to reach a conclusion
  
• Axioms (postulates) fundamental assumptions
• Rules of Inference Valid means to draw
  conclusions from one or more assertions
  
• Conjecture Statement neither proved or disproved

4
Rules of Inference

• Modus Ponens (law of detachment)
• If p is true, and if the implication if p, then
  q is also true, then q is true
  
• p
• p ? q
• ? q

5
Modus Ponens

• If I have enough money, I will go to the movies
• I have enough money
• If you drive too fast, you will get hurt
• You drive too fast

6
Rules of Inference

• Addition
• If I am wearing a shirt, then I am wearing a
  shirt or tie.
  
• Simplification
• If I am wearing a shirt and tie, then I am
  wearing a tie
  
• Conjunction
• I am wearing a tie. I am wearing a shirt. I am
  therefore wearing a shirt and tie
  
• Modus Ponens
• Modus tollens
• I am not wearing socks. If I am wearing shoes,
  then I am wearing socks. Therefore, I am not
  wearing shoes.
  

7
Rules of Inference

• Hypothetical syllogism
• If I am wearing a shirt, then I am wearing a tie.
  If I am wearing a tie, then I am wearing a
  tiepin. Therefore, if I am wearing a shirt, I am
  wearing a tiepin
• Disjunctive syllogism
• I am wearing a sombrero or a beret. I am not
  wearing a beret. Therefore, I am wearing a
  sombrero.
  
• Resolution
• I am going to the beach or to the movies. I am
  not going to the beach or I am going to Chicago.
  Therefore, I am going to the movies or to Chicago.

8
Conclusions based on inference rules

• It is snowing. It is windy
• Conclusion
• Rule
• I have a turkey sandwich or a ham sandwich. I
  do not have a turkey sandwich.
  
• Conclusion
• Rule
• If you finish on time, I will pay you 500. I do
  not pay you 500.
  
• Conclusion
• Rule
• I have a corvette or a mustang. I have a beetle
  or I do not have a mustang.
  
• Conclusion
• Rule

9
Valid Arguments

• Joe works at Menards or Pier 1 Imports
• If Joe is not an interior decorator, then he does
  not work at Pier 1 Imports.
  
• If Joe is an interior decorator, then he needs to
  buy decorations.
  
• Joe does not work at Menards
• Therefore, Joe needs to buy decorations

10
Valid Arguments

• If it does not rain or if it is not foggy, then
  the sailing race will be held and the lifesaving
  demonstration will go on.
  
• If the sailing race is held, then the trophy will
  be awarded
  
• The trophy was not awarded
• It rained.

11
Conclusions (in-class)

• If I lift weights, then I am sore the next day.
• I use the hot tub when I am sore.
• I did not use the hot tub.
• Prove I am not sore.
• I did not lift weights yesterday

12
Fallacies

• Affirming the conclusion
• If I run too fast, then I am out of breath.
• I am out of breath.
• ? I ran too fast
• Denying the hypothesis
• If I order early, I will get my meal hot.
• I didnt order early.
• ? I will get a cold meal.

13
Fallacies

• If n is a real number such that n 1, then n2
  1. Suppose that n2 1. Then n 1.
  
• The square root of an irrational number is
  irrational. Let x p2, which is irrational.
  Therefore, x p is irrational.

14
Quantifiers and Inference

• Universal Instantiation
• Given that ?xP(x), P(C) if C is a member of the
  universe of discourse
  
• Universal Generalization
• ?xP(x) is true if we show that P(c) is true for
  arbitrary c
  
• Existential Instantiation
• If ?xP(x), then we can conclude that P(c) is true
  for some c
  
• Existential Generalization
• If we find a c such that P(c), then ?xP(x)

15
Quantified Conclusions

• Every student in ECE 357 is a sophomore
• Every sophomore in the class passed the final
• Jim Zlogar is a student in ECE357.
• Someone in this room left a wrapper on the
  floor.
  
• Everyone in this room is neat and tidy.
• Someone who is neat and tidy left a wrapper on
  the floor.

16
Proofs

• Direct Proof
• Prove p?q, show that if p is true, q must be true
  (no combination of p true and q false)
  
• Indirect Proof
• Use direct proof on contrapositive
• Vacuous Proof
• Show that p is always false
• Trivial Proof
• Show that q is always true

17
Direct Proof

• The square of an even number is an even number

18
Indirect Proof

• If n is an integer and n3 5 is odd, then n is
  even

19
Vacuous Trivial Proofs

• Show P(1) is true for the propositional function
  
  
• x2
• Show P(0) is true for the propositional
  function
  
• If x

20
Proof by Contradiction

• To prove a proposition p, begin w/ the assumption
  ?p
  
• Find a contradiction q such that ?p ? q
• Contradiction means that q False
• If ?p ? q is the case, and if q is always false,
  then ?p must also be false
  
• Therefore, p is true
• N.B. these proofs can be very tricky

21
Proof by Contradiction

• Prove If (7n 4) is odd, then n is odd

22
Proof by Contradiction

• Prove If (a ? 0), then (ax b 0) has exactly
  one solution for x, given real numbers a,b

23
Additional Proofs

• Proof by cases
• To prove (p1 ? p2 ? p3 ? ? pn) ? q show that pi
  ? q for all I
  
• Proofs of equivalence
• To show (p ? q), prove (p?q) ? (q?p)
• Existence Proofs
• ?xP(x) is proved by finding an element P(c) ?
  True
  
• Counterexample
• Prove ?xP(x) false by finding P(c) ? False