1. Lecture 5 CS 1813 – Discrete Mathematics
Lecture 5 - CS 1813 Discrete Math, University of Oklahoma
2/22/2019
Lecture 5 CS 1813 – Discrete Mathematics
What Not
and
Rules Galore
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. Or Introduction and Or Elimination
Theorem (Or Commutes) a  b |– b  a a
{IL}
a  b
Or Intro Left b
{IR}
a  b
Or Intro Right
a  b a |– c b |– c
{E} c
Or Elimination
Suppose
We could prove the formula b  a assuming a alone
And we could prove b  a again, assuming b alone
Then we could apply E to prove b  a assuming a  b
assumptions temporarily admitted
discharged
by E a
{IR}
b  a b
{IL}
b  a
remaining assumption
a  b
{E}
b  a
conclusion
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. CS 1813 Discrete Mathematics, Univ Oklahoma
What is Not?
Definition of the Not Operator
a means aFalse
Rationale
You don’t want False to follow from True
So, if aFalse is True, then a better be False
Which means, a must be True
True means False - Right?
So, True means FalseFalse (def’n of True)
Consistent with definition of the Not operator
All this stuff is rigged for consistency
Countless logicians worked it out a long time ago
Proofs of WFFs containing Not use this def’n
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

4. CS 1813 Discrete Mathematics, Univ Oklahoma
Modus Tollens
Theorem (Modus Tollens)
a  b, b |– a
Proof
This is a homework problem
Proved in text, too
Your job
Express in natural deduction form
Convert to proof-checker notation
We will take it as a proven theorem
Counting on you to succeed
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. Inference Rules Galore
Theorems become new inference rules
thm cited on right
(like a rule)
premises above
the line
a  b b
{modTol} a
conclusion below
a  b
{Comm}
b  a
a  b
{Comm}
b  a
ab bc
{Chain}
ac
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. Or Commutes Inside Not (a  b) |– (b  a)
Plan
Prove b  a  a  b
Cite Modus Tollens
a  b, b |– a
(a  b)
{()Comm}
(b  a)
Not Or Commutes as rule
assumption temporarily admitted
Using Or-Commutes Theorem
discharged
by I
b  a
{Comm}
a  b
Using Modus Tollens
{I}
b  a  a  b
(a  b)
{modTol}
remaining assumption
(b  a)
conclusion
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. Using Theorems as Inference Rules in the Proof Checker
andCommThm = Theorem [A `And` B] (B `And` A)
andComm = Use andCommThm
Prototype Citation — [Proof] `andComm` Prop
conclusion here
premises go here
infix citation
Just like ordinary rule-citation
Except premises are in a sequence instead of a tuple
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8. CS 1813 Discrete Mathematics, Univ Oklahoma
Curried Constructors
Use - another constructor for the type Proof
Use :: Theorem -> [Proof ] -> Prop -> Proof three arguments
Familiar constructors for the type Proof
AndI :: (Proof, Proof) -> Prop -> Proof two arguments
AndEL :: Proof -> Prop -> Proof st premises
ImpI :: Proof -> Prop -> Proof nd conclusion
How many arguments do these require?
Use (Theorem [A `And` B] (B `And` A))
Use andCommThm
where andCommThm = Theorem [A `And` B] (B `And` A)
Two arguments remain to be supplied
First argument is specified (the theorem being cited)
That leaves two other arguments
Premises and conclusion remain to be supplied by citation
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. Preparing Theorems for Citation as Rules
define module in file named Lecture4Thms.hs
import Lecture4Thms (to use def’ns elsewhere)
module Lecture4Thms where
andCommThm = Theorem [A `And` B] (B `And` A)
andComm = Use andCommThm
name for theorem
name for citing theorem as inference rule
orCommThm = Theorem [A `Or` B] (B `Or` A)
orComm = Use orCommThm
impChainThm = Theorem [A `Imp` B, B `Imp` C] (A `Imp` C)
impChain = Use impChainThm
import Lecture4Thms -- to cite theorems from Lecture4Thms.hs
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

10. Or-Commutes-In-Not Theorem and Proof in Proof-Checker Notation
Lecture 5 - CS 1813 Discrete Math, University of Oklahoma
2/22/2019
Or-Commutes-In-Not Theorem and Proof in Proof-Checker Notation
orCommInNotFwdThm = Theorem [Not(A `Or` B)] (Not(B `Or` A))
proofOrCommInNotFwdThm =
[ ( [ Assume(B `Or` A) ]
{ }`orComm`
(A `Or` B))
{ }`ImpI`
((B `Or` A) `Imp` (A `Or` B)) , Assume(Not(A `Or` B)) ]
{ }`modTol`
(Not(B `Or` A))
Premise sequence for orComm
[Proof]
Assumption discharged
by ImpI
premises for modTol
[Proof, Proof]
At this point, again go to the Haskell session with lecture04.hs and again demonstrate use of proof checker.
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

11. Not Or Elimination – Left (a  b) |– a
Plan –
Assume a
Derive False
Cite {I}
(a  b)
{()EL} a
Not Or Elimination (Left) as rule
assumption temporarily admitted
means (a  b)
(assumption of theorem)
discharged by I a
{IL}
a  b
Cite what rule?
(a  b)  False
{E}
False
Now what?
{I}
a  False
means a
(conclusion of theorem)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

12. Not Or Elimination – Right (a  b) |– b
{()ER} b
Not Or Elimination (Right) as rule
Plan –
Flip it around with {()Comm}
Cite {()EL}
(a  b)
{()Comm}
(b  a)
Cite what rule?
{()EL} b
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

13. DeMorgan’s Law for Or (Forward Half) (a  b) |– (a)  (b)
{DeMF}
(a)(b)
DeMorgan Or Fwd as rule
Plan
Derive halves of conclusion separately
Cite {I}
(a  b)
{()EL} a
(a  b)
{()ER} b
{I}
(a)  (b)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

14. DeMorgan’s Law for Or (Backward Half) (a)  (b) |– (a  b)
Plan
Use {E} to derive False from a  b
Cite {I} to conclude (a  b)False
Hold onto your seat
This proof is supercharged
prove this as an exercise
Admit for planned {E}
Admit for planned {I}
(a)  (b)
(a)  (b)
{EL} a
{ER} b
Discharged by {I}
a  b
Discharged by {E} a b
{ +&- }
{ +&- }

False

False
{E}
False
{I}
(a  b)  False
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

15. Rules Galore! — Theorems in Rule Form
(a  b)
{DeMF}
(a)(b)
DeMorgan Or Fwd
(a)(b)
{DeMB}
(a  b)
DeMorgan Or Bkw
a  b
{Comm}
b  a
And Commutes
a  b
{Comm}
b  a
Or Commutes
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 a
{ +&- }
False
NeverBoth
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

16. CS 1813 Discrete Mathematics, Univ Oklahoma
What Is Discrete Math?
discrete math
An area of mathematics dealing with countable collections of entities
Example Topics
Logic Trees and graphs
Functions and relations Grammars
Combinatorics Number theory
et cetera
discreet math
A math that shows good judgment in conduct
Not covered in discrete math
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

17. CS 1813 Discrete Mathematics, Univ Oklahoma
End of Lecture
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page