CSC195, Class 6: Boolean Proof

Summary:
* Review: Why prove?
* Laws and Rules
* Twelve key laws
* Rule of subsititution
* Rule of transitivity

Notes:
* I hate 
  + paperwork
  + one-hour meetings that take two hours
* Homework 2 due.
* Homework 3 assigned.
* Goal
* Sam's six+ P's 
  http://glimmer.cs.grinnell.edu/Scheme/pdoc.html
* Javadoc

/**
 * The value true.
 */
public class True {
class definitions
field definitions
method definitions
constructor definitions

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Comment: We will want to prove things in Boolean logic
What? This assertion holds (always)
Why?  Put checks in our program
      + Things the computer checks
      + Things you check

Techniques for proving that a proposition is a tautology
* Proof by truth table
* Prove equivalent to a simpler expression and prove
  that that expression is a tautology

We can make big improvements by "simplifying" expressions

Incorrect theorem
  ((a or b or c or d or (not a) or (not b)or (not c)) equals d)


Theorem
  ((a + b + c + d + (-a) + (-b) + (-c)) = d
Proof

  ((a + (-a) + b + (-b) + c + (-c) + d) = d
				By the extended commutative 
				property of addition

  (0 + 0 + 0 + d) = d
				By the rule sometimes known as
				addition of inverses

  d = d
				By the additive identity property

  Q.E.D.

1. Commutativity

  (E1 v E2)  =  (E2 v E1)
  (E1 ^ E2)  =  (E2 ^ E1)
  (E1 = E2)  =  (E2 = E1)
  (E1 => E2) =  (E2 => E1)  FALSE

How do we prove these laws?  Truth table.
How do we disprove the the fallacy?
 Let E1 by false, Let E2 be true
  (false => true) = (true => false)
  true = (true => false)
  true = false
  false

Erik's claim


  (E1 => E2) = ((not E2) => (not E1))
  Let E1 be false, E2 be true

  (false => true) = ((not true) => (not false))

2. "Excluded middle"

E or (not E) = True

3. Law of contradiction

E and (not E) = False

4. Law of "additive identity"

E or false = E
false or E = E

5. Law of "multiplicative identity"

E and true = E
true and E = E

6. Distributive property (of multiplication across addition)

a * (b + c) = (a * b) + (a * c)

a ^ (b v c) = (a ^ b) v (a ^ c)

a v (b ^ c) = (a v b) ^ (a v c)

7. De Morgan's laws
not (b v c) = (not b) ^ (not c)

not (b ^ c) = (not b) v (not c)

8. Associativity

(a v b) v c = a v (b v c)
(a ^ b) ^ c = a ^ (b ^ c)
((a = b) = c) = (a = (b = c))

9. Law of negation

(not (not a)) = a

10. Law of implication

(E1 = E2) = ((E1 => E2) and (E2 => E1))

Others in the book.

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Two rules (different than laws)

What's the differnce between a rule and a law (at least to
Gries, who is careful in his terminology)?

Rule of substitution

If E1 = E2 and P is a proposition involving variable p

Then "P with E1 substituted for p" =
     "P with E2 substituted for p"

Proposition: (foo and bar)
  (not (not a)) = a

((not (not a)) and bar) = (a and bar)

a = 0

p++

a++ = 0++

Rule of transitivity

If A = B and B = C then A = C