Proofs

1
Proofs
2
Mathematical System
definitions
used to create terms
Rules of Inference
axioms
assumed to be true
Logic is a tool for the analysis of inference
theorems
proved to be true
3
Proofs

• Rules of Inference
• Formal proofs
• Methods of Proofs
• Mathematical induction

4
Inference

• Inference / deductive reasoning is the process
  of drawing valid conclusions from a sequence of
  propositions
• (p1, p2, p3, pn) ? q

p1 p2 p3 pn ?q
propositions / hypotheses
conclusion
5
Inference Example
p p?q ?q

• Given p is true and p?q is true, what can be
  inferred?
• Example
• If I eat breakfast, I will not be hungry.
• I eat breakfast.
• Therefore, I will not be hungry.

6
Rules of Inference

• The Rules of Inference derive from some well
  known tautologies. For example, modus ponens is
  obtained from the tautology
• (p ? (p ? q)) ? q

p p?q ?q
Modus ponens
7
Inference Example
q p?q ?p

• Given q is true and p?q is true, what can be
  inferred?
• Example
• I did not bet on the game.
• If the Phillies lost, then I bet on the game.
• Therefore, the Phillies won!

Modus tollens
8
Rules of Inference
9
Rules of Inference (example)

• Problem
• Randy works hard. If Randy works hard, then he
  is a dull boy. And if Randy is a dull boy, then
  he will not get the job. Therefore, Randy will
  not get the job.
• Solution
• p Randy works hard
• q Randy is a dull boy
• r Randy will get the job
• p
• p ? q
• q ? r
• ? r

10
Using Rules of Inference

• Problem
• If you send me an email message, then I will
  finish writing the program. If I finish writing
  my program, then I will go to sleep early. You
  sent me an email. Therefore, I went to sleep
  early.
• Solution
• p you send me an email message
• q I will finish writing the program
• r I will go to sleep early
• p ? q
• q ? r
• p
• ? r

11
Rules of Inference for Quantified Statements
12
Using Rules of Inference for Quantified Statements

• Problem
• Every student in discrete mathematics has taken
  a course in computer science. And it is true that
  Marla is a student in this class. Therefore,
  Marla has taken a course in computer science.
• Solution
• P(x) x is a student in the discrete mathematics
  class
• Q(x) x has taken a course in computer science
• ?x ( P(x) ? Q(x) )
• P(Marla)
• ? Q(Marla)

13
Homework

• Read Section 1.5
• Exercises 4, 6, 14