Logical Equivalences

```Discussion #10
Logical Equivalences
Discussion #10
1/16
Topics
•
•
•
•
Laws
Duals
Manipulations / simplifications
Normal forms
– Definitions
– Algebraic manipulation
– Converting truth functions to logic expressions
Discussion #10
2/16
Laws of , , and 
Law
P  P  T
P  P  F
PFP
PTP
PTT
PFF
PPP
PPP
(P)  P
Discussion #10
Name
Excluded middle law
Identity laws
Domination laws
Idempotent laws
Double-negation law
3/16
Law
PQQP
PQQP
(P  Q)  R  P  (Q  R)
(P  Q)  R  P  (Q  R)
(P  Q)  (P  R)  P  (Q  R)
(P  Q)  (P  R)  P  (Q  R)
(P  Q)  P  Q
(P  Q)  P  Q
P  (P  Q)  P
P  (P  Q)  P
Discussion #10
Name
Commutative laws
Associative laws
Distributive laws
De Morgan’s laws
Absorption laws
4/16
Can prove all laws by truth tables…
P
Q
 (P  Q)

P  Q
T
T
F
T
T
F
F F
T
F
T
F
T
F
T T
F
T
T
F
T
T
T F
F
F
T
F
T
T
T T
De Morgan’s law holds.
Discussion #10
5/16
Absorption Laws
P  (P  Q)  P
Venn diagram proof …
P  (P  Q)  P
P
Q
Prove algebraically …
P  (P  Q)  (P  T)  (P  Q)
Discussion #10
identity
 P  (T  Q)
distributive (factor)
PT
domination
P
identity
6/16
Duals
• To create the dual of a logical expression
1) swap propositional constants T and F, and
2) swap connective operators  and .
P  P  T




P  P  F
Excluded Middle
• The dual of a law is always a law!
• Thus, most laws come in pairs  pairs of duals.
Discussion #10
7/16
Why Duals of Laws are Always Laws
We can always do the following:
P  P  T
Negate both sides
(P  P)  T
Apply De Morgan’s law
P  P  T
Simplify negations
Since a law is a tautology,
substitute X for X
Simplify negations
Discussion #10
P  P  F
(P )  (P )  F
P  P  F
8/16
Normal Forms
• Normal forms are standard forms, sometimes
called canonical or accepted forms.
• A logical expression is said to be in disjunctive
normal form (DNF) if it is written as a disjunction,
in which all terms are conjunctions of literals.
• Similarly, a logical expression is said to be in
conjunctive normal form (CNF) if it is written as a
conjunction of disjunctions of literals.
Discussion #10
9/16
DNF and CNF
• Disjunctive Normal Form (DNF)
( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
Term
Examples:
Literal, i.e. P or P
(P  Q)  (P  Q)
P  (Q  R)
• Conjunctive Normal Form (CNF)
( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
Examples:
Discussion #10
(P  Q)  (P  Q)
P  (Q  R)
10/16
Converting Expressions
to DNF or CNF
The following procedure converts an expression to DNF or CNF:
1. Remove all  and .
2. Move  inside. (Use De Morgan’s law.)
3. Use distributive laws to get proper form.
Simplify as you go. (e.g. double-neg., idemp., comm., assoc.)
Discussion #10
11/16
CNF Conversion Example
( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
((P  Q)  R  (P  Q))
 ((P  Q)  R  (P  Q))
 (P  Q)  R  (P  Q)
 (P  Q)  R  (P  Q)
(DNF)  (P  Q)  R  (P  Q)
 ((P  R)  (Q  R))  (P  Q)
 ((P  R)  (P  Q)) 
((Q  R)  (P  Q))
 (((P  R)  P)  ((P  R)  Q)) 
(((Q  R)  P)  ((Q  R)  Q))
 (P  R)  (P  R  Q)  (Q  R)
Discussion #10
impl.
deM.
deM.
double neg.
distr.
distr.
distr.
assoc. comm. idemp.
12/16
CNF Conversion Example
( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
((P  Q)  R  (P  Q))
 ((P  Q)  R  (P  Q))
 (P  Q)  R  (P  Q)
 (P  Q)  R  (P  Q)
(DNF)  (P  Q)  R  (P  Q)
 ((P  R)  (Q  R))  (P  Q)
 ((P  R)  (P  Q)) 
((Q  CNF
R)  (P  Q))
Using
the 
commutative
idempotent
 (((P
R)  P) and
 ((P
 R)  Q)) 
laws on the previous step and then the
(((Q  R)  P)  ((Q  R)  Q))
distributive law, we obtain this formula
as(Pthe
 conjunctive
R)  (P normal
 R 
Q)  (Q  R)
form.
Discussion #10
impl.
deM.
deM.
double neg.
distr.
distr.
distr.
assoc. comm. idemp.
13/16
CNF Conversion Example
( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
(P  R)  (P  R  Q)
((P  Q)  R  (P  Q))
 (Q  R)
 ((P  Q)  R  (P  Q))
impl.
 (P  R)  (P  R  Q)
 (P  Q)  R  (P  Q)
 (F  Q deM.
R) - ident.
 (P  Q)  R  (P  Q)
 (P  R)  ((PdeM.
 F)
double
neg. distr.
(DNF)  (P  Q)  R  (P  Q)
 (Q  R))
- comm.,
 ((P  R)  (Q  R))  (P Q)
(P  R)  (Fdistr.
 (Q  R))
 ((P  R)  (P  Q)) 
distr.- dominat.
 (P  R)  (Q  R) - ident.
((Q  R)  (P  Q))
 (((P  R)  P)  ((P  R)  Q)) 
distr.
(((Q  R)  P)  ((Q  R)  Q))
 (P  R)  (P  R  Q)  (Q  R)
assoc. comm. idemp.
Discussion #10
14/16
DNF Expression Generation
P
Q
R

T
T
T
F
T
T
F
T
T
F
T
T
T
F
F
F
F
T
T
F
F
T
F
F
F
F
T
T
F
F
F
F
The only definition of
 is the truth table
(P  Q  R)
(P  Q  R)
minterms
(P  Q  R)
  (P  Q  R)  (P  Q  R)  (P  Q  R)
Discussion #10
15/16
CNF Expression Generation
1.
2.
3.
Find .
Find the DNF of .
Then, use De Morgan’s law to get the
CNF of  (i.e. ()  )
P
Q


T
T
T
F
T
F
F
T
F
T
T
F
F
F
F
T
}
Form a
conjunction of
max terms
max terms
(P  Q)
(P  Q)
(P  Q)
(P  Q)
  (P  Q)  (P  Q) DNF of 
  f  ((P  Q)  (P  Q))
 (P  Q)  (P  Q) De Morgan’s
 (P  Q)  (P  Q) De Morgan’s, double neg.
Discussion #10
16/16
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