1. CS 1813 – Discrete Mathematics
Lecture 10 - CS 1813 Discrete Math, University of Oklahoma
11/22/2018
CS 1813 – Discrete Mathematics
Review of
Propositional Calculus
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. Truth Tables for Logical Operators
P Q P  Q
False False False
False True False
True False False
True True True
P  Q
False
True
P  Q
PQ
PQ
 P
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. Semantic Reasoning with Truth Tables
Proposition (WFF): ((P  Q)((P)Q))
P Q
False False
False True
True False
True True
(P  Q)
(P)
((P)Q)
((PQ)((P)Q))
Some True: prop is Satisfiable
False
True
False
False
True
True
True
True
True
False
True
True
All False: Contradiction
(not satisfiable)
If prop is True when all variables areTrue:
P, Q ((PQ)((P)Q))
True
False
If they were all True: Tautology
True
True
Formal statement about meaning
Proved via truth table
double turnstile
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

4. Lecture 10 - CS 1813 Discrete Math, University of Oklahoma
11/22/2018
Rules of Inference
Discrete Mathematics with a Computer Springer, 2000
Fig 2.1, Hall/O’Donnell
After presenting this slide, put up a copy on the overhead projector to refer to in upcoming proofs.
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. Some Intrinsic Data Structures in Haskell
Lecture 10 - CS 1813 Discrete Math, University of Oklahoma
11/22/2018
Some Intrinsic Data Structures in Haskell
Sequences (aka lists — in primitive PLs, done as linked lists)
[x1, x2, …] all x’s must have same type
Examples: [t] means a sequence with elements of type t
[1, 9, 3, 27] type: [Integer]
[And A B, Or P Q, B] type: [Prop]
Tuples (like structs or records in other programming languages)
(c1, c2) pair – components may have different types
(c1, c2 , c3) 3-tuple (longer tuples OK—must be at least 2 components)
Examples: (t1, t2) means a pair where component k has type tk
(7, And A B) type: (Integer, Prop)
(AndEL (Assume(A `And` B)) A, Assume B) type: (Proof, Proof)
Constructors are things that build data of the type in which they appear as part of the definition.
Proof
Proof
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. Some Theorems in Rule Form
a  b
{Comm}
b  a
And Commutes
a  b
{Comm}
b  a
Or Commutes
a a
{ +&- }
False
NeverBoth
ab bc
{Chain}
ac
Implication Chain Rule
(a  b)
{()Comm}
(b  a)
Not Or Commutes
a  b b
{modTol} a
Modus Tollens
a  b
{conPosF}
(b)(a)
Contrapositive Fwd
{noMiddle}
a  (a)
Law of Excluded Middle
a  b
{F}
(a)  b
Implication Fwd
(a)  b
{F}
a  b
Implication Bkw
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. More Theorems in Rule Form
(a  b)
{DeMF}
(a)(b)
DeMorgan Or Fwd
(a)(b)
{DeMB}
(a  b)
DeMorgan Or Bkw
(a  b)
{DeMF}
(a)(b)
DeMorgan And Fwd
(a)(b)
{DeMB}
(a  b)
DeMorgan And Bkw
a  b a
{disjSyll} b
Disjunctive Syllogism
(a)
{ F} a
Double Negation Fwd a
{ B}
(a)
Double Negation Bkw
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8. Laws of Boolean Algebra page 1
From Fig 2.1, Hall & O’Donnell, Discrete Math with a Computer, Springer, 2000
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. Laws of Boolean Algebra page 2
From Fig 2.1, Hall & O’Donnell, Discrete Math with a Computer, Springer, 2000
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

10. Some Theorems in Boolean Algebra
(a  b)  b = b { absorption}
(a  b)  b = b { absorption}
(a  b)  c = (a  c)  (b  c) { imp}
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

11. Boolean Equations and Tautologies
Tautology
WFF that is true for all combinations of values of its atomic consituents
Let p, q stand for arbitrary WFFs
Suppose Boolean laws prove p = q
Then the following WFFs are tautologies
p  q
q  p
p  q
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

12. Theorems and Tautologies
Let p, q, … stand for arbitrary WFFs
p |– q
If this is a theorem, then p  q is a tautology
p, q, r |– s
If this is a theorem, then (p  q  r)  s is a tautology
|– p
If this is a theorem, then p is a tautology
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

13. CS 1813 Discrete Mathematics, Univ Oklahoma
What It All Means
Notions of Consistency in Formal Systems
If a |– b, then a |= b (a, b arbitrary WFFs)
Completeness in Formal Systems
If a |= b, then a |– b (a, b arbitrary WFFs)
If b is true whenever a is, there is a proof of a |– b
Propositional Logic
Consistency — propositional logic is consistent
Inference preserves tautologies
Inconsistency would make all WFFs tautologies
Some WFFs aren’t tautologies — QED (consistency)
Completeness — propositional logic is complete
That’s no surprise, is it?
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

14. Kurt Gödel — the first computer scientist
Arithmetic is consistent
comforting, but not surprising
Arithmetic is not complete
Some statements about numbers are consistent with all the axioms, but cannot be proved
A humongous surprise for mathematicians in the 1930s
Consistency and Completeness
Impossible goal
No formal system can be both
Except stripped-down systems (less powerful than arithmetic)
In essence, this has nothing to do with numbers
It has to do with the limits of formal systems (computers)
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I — Gödel, 1931
46 definitions + many lemmas precede main point
Translation by Meltzer — $6.25, Amazon.com
The recognition of Kurt Gödel as the first computer scientist is an insight of John Allen, a present-day logician and computer scientist, author of Anatomy of Lisp.
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

15. Practice Problem (a)  b |– a  b Implication Backward
Lecture 10 - CS 1813 Discrete Math, University of Oklahoma
11/22/2018
Practice Problem (a)  b |– a  b Implication Backward a
{ B}
(a)  b (a)
{disjSyll} b
{I}
a  b
Form teams of 5 students each.
Give them the following practice problems, one at a time.
Give them 8 to 10 minutes to work on a problem.
Ask a successful team to present it’s solution to the class.
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

16. Practice Problem (a)  (b) |– b  a contrapositive backwards
{conPosF} { F} { F}
(b)  (a) b a
{F} {IL} {IR}
((b))  (a) (b)  a (b)  a
{E}
(b)  a
{B}
b  a
That’s ugly! There must be a better way.
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

17. Practice Problem (a)  (b) |– b  a contrapositive backwards
Better way b
{ B}
(a)  (b) (b)
{modTol}
(a)
{ F} a
{I}
b  a
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

18. Practice Problem a  ((a)  b) = a  b
= (a  (a))  (a  b) { dist/}
= False  (a  b) { comp}
= (a  b)  False { commutes}
= a  b { id}
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

19. Lecture 10 - CS 1813 Discrete Math, University of Oklahoma
11/22/2018
The End
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page