101_Lecture 7 CS 1813 – Discrete Mathematics

Lecture 7
CS 1813 – Discrete Mathematics

Equational Reasoning
Back to the Future: High-School Algebra

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

1


Some Laws of Algebra
 a+0=a

{+ identity}

 (-a) + a = 0

{+ complement}

 a× 1=a

{× identity}

 a× 0=0

{× null}

 a+b=b+a

{+ commutative}

 a + (b+c) = (a+b) + c

{+ associative}

 a× (b+c) = a× b + a× c

{distributive law}

Equations go both ways

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

2


Theorem

(-1) × (-1) = 1

(-1) × (-1)
= ((-1) × (-1)) + 0 {+ id}
= ((-1) × (-1)) + ((-1) + 1)
{+ comp}
= (((-1)× (-1)) + (-1)) + 1 {+ assoc}
= (((-1)× (-1)) + (-1)× 1) + 1

{× id}
= ((-1)× ((-1) + 1)) + 1
{dist law}
= ((-1)× 0) + 1 {+ comp}
= 0 + 1 {× null}
= 1 + 0 {+ comm}
= 1 {+ id}

QED
proof by equational reasoning
CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

3


Laws of Boolean Algebra
From Fig 2.1, Hall & O’Donnell, Discrete Math with a Computer,
Springer, 2000

page 1

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

4

Laws of Boolean Algebra
From Fig 2.1, Hall & O’Donnell, Discrete Math with a Computer,
Springer, 2000

page 2

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

5


Theorem (a ∧ False) ∨ (b ∧ True) =
b
equations

{rule} substitution
[formula in eqn / variable in

rule]

(p ∧ False) ∨ (q ∧ True)

names changed to clarify

substitutions

=
=

=
=

False ∨ (q ∧ True)
(q ∧ True) ∨ False
q ∧ True
q

{∧ null}

[p /a]

{∨ comm} [False /a] [q∧True /b]
{∨ id} [q ∧ True /a]
{∧ id} [q /a]

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

QED

6


Using
ng
i
t
r

o
p
m
I
ls
o
o
t

Equational Proof Checker

import Stdm
th7 = (P `And` FALSE) `Or`
( Q `And` TRUE)
`thmEq` Q
pr7 =
startProof ((P `And` FALSE) `Or`
(Q `And` TRUE))
<-> (FALSE `Or` (Q `And` TRUE),
andNull)
<-> ((Q `And` TRUE) `Or` FALSE,
orComm)
<-> (Q `And` TRUE,
orID)
<-> (Q,
andID)

Notepad window

Gr

in d e e n
fro ica
m te s
u
Prelude> :cd DMf00se cmd
r

Prelude> :cd Lectures
Prelude> :load lecture07.hs
Reading file "lecture07.hs":
Reading file "Stdm.lhs":
Reading file "lecture07.hs":
Hugs session for:
C:\HUGS98\lib\Prelude.hs
Stdm.lhs
lecture07.hs
Main> check_equation th7 pr7
The proof is correct

Hugs Session

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

7


Equations the Proof Checker
Knows

andNull
orNull
andID
orID
andIdempotent
orIdempotent
andComm
orComm
andAssoc
orAssoc
andDistOverOr
orDistOverAnd

deMorgansLawAnd
deMorgansLawOr
negTrue
negFalse
andCompl
orCompl
dblNeg
currying
implication
contrapositive
absurdity

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

8


Theorem

(a ∧ b) ∨ b = b

∨ absorption

equations

{rule} substitution
[formula in eqn / variable in

rule]

(p ∧ q) ∨ q

names changed to clarify

substitutions

=
=
=
=
=

(p ∧ q) ∨ (q ∧ True)
(q ∧ p) ∨ (q ∧ True)

q ∧ (p ∨ True)
q ∧ True
q

{∧ id} [q /a]
{∧ comm} [p /a] [q /b]
{∧ dist over ∨} [q /a] [True /b] [p /c]
{∨ null} [p /a]
{∧ id} [q/a]
QED
CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

9


Theorem

(a ∨ b) ∧ b = b

∧ absorption

equations

{rule} substitution
[formula in eqn / variable in

rule]

(p ∨ q) ∧ q

names changed to clarify

substitutions

… exercise …
=q

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

10


Consistent, But Not Minimal
redundancy among laws of Boolean
algebra

Deriving the contrapositive law
Theorem (contrapositive): a → b = ¬b → ¬a
A proof using laws other than the contrapositive law
equations

{rule} substitution
[formula in eqn / variable in rule]

p →q
= (¬p) ∨ q

{imp} [p /a] [q /b]
= ¬(¬((¬p) ∨ q))
{dbl neg} [(¬p) ∨ q /a]
= ¬((¬(¬p)) ∧ (¬q))
{DeMorgan ∨} [¬p /a] [q /b]
= ¬(p ∧ (¬q))
{dbl neg} [p /a]
= (¬p) ∨ (¬(¬q))
{DeMorgan ∧} [p /a] [¬q /b]
= (¬(¬q)) ∨ (¬p)
! {∨ comm} [¬p /a] [¬(¬(q)) /b]
y
p
op ut?{imp} [¬q /a] [¬p /b]
l
s
= (¬q) → (¬p)
f is hortc
o
o
pr the s
QED
s
i
Th re’s
CS 1813 Discrete Mathematics, Univ
e
h
W
Oklahoma

Copyright © 2000 by Rex Page

11


End of Lecture 7

CS 1813 Discrete Mathematics, Univ
Oklahoma
Copyright © 2000 by Rex Page

12