1. Discussion #10 Logical Equivalences
Chapter 1, Section 5
Discussion #10

2. Topics Laws Duals Manipulations / simplifications Normal forms
Definitions
Algebraic manipulation
Converting truth functions to logic expressions
Chapter 1, Section 5
Discussion #10

3. Laws of , , and  Name Law Excluded middle law Contradiction law
P  P  T
P  P  F
Identity laws
P  F  P
P  T  P
Domination laws
P  T  T
P  F  F
Idempotent laws
P  P  P
P  P  P
Double-negation law
(P)  P
Chapter 1, Section 5
Discussion #10

4. Law Name Commutative laws P  Q  Q  P P  Q  Q  P Associative laws
(P  Q)  R  P  (Q  R)
(P  Q)  R  P  (Q  R)
Distributive laws
(P  Q)  (P  R)  P  (Q  R)
(P  Q)  (P  R)  P  (Q  R)
De Morgan’s laws
(P  Q)  P  Q
(P  Q)  P  Q
Absorption laws
P  (P  Q)  P
P  (P  Q)  P
Chapter 1, Section 5
Discussion #10

5. Can prove all laws by truth tables…F T Q  P 
(P  Q)  Q P T F F T T T F T F T F
De Morgan’s law holds.
Chapter 1, Section 5
Discussion #10

6. Absorption Laws P  (P  Q)  P P  (P  Q)  P Prove algebraically …
Venn diagram proof …
P  (P  Q)  P
P Q
Prove algebraically …
P  (P  Q)  (P  T)  (P  Q) identity
 P  (T  Q) distributive (factor)
 P  T domination
 P identity
Chapter 1, Section 5
Discussion #10

7. 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 Excluded Middle
 
 
P  P  F Contradiction
The dual of a law is always a law!
Thus, most laws come in pairs  pairs of duals.
Chapter 1, Section 5
Discussion #10

8. Why Duals of Laws are Always Laws
We can always do the following:
Start with law P  P  T
Negate both sides (P  P)  T
Apply De Morgan’s law P  P  T
Remove double negatives P  P  F
Since a law is a tautology, (P )  (P )  F
substitute X for X
Remove double negatives P  P  F
Chapter 1, Section 5
Discussion #10

9. 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.
Chapter 1, Section 5
Discussion #10

10. DNF and CNF Conjunctive Normal Form (CNF)
Disjunctive Normal Form (DNF)
( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
Term
Literal, i.e. P or P
Examples: (P  Q)  (P  Q)
P  (Q  R)
Conjunctive Normal Form (CNF)
( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
Examples: (P  Q)  (P  Q)
P  (Q  R)
Chapter 1, Section 5
Discussion #10

11. Converting Expressions to DNF or CNF
The following procedure converts an expression to DNF or CNF:
Remove all  and .
Move  inside. (Use De Morgan’s law.)
Use distributive laws to get proper form.
Simplify as you go. (e.g. double-neg., idemp., comm., assoc.)
Chapter 1, Section 5
Discussion #10

12. CNF Conversion Example ( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
((P  Q)  R  (P  Q))  ((P  Q)  R  (P  Q)) impl.  (P  Q)  R  (P  Q) deM.  (P  Q)  R  (P  Q) deM.  (P  Q)  R  (P  Q) double neg.  ((P  R)  (Q  R))  (P  Q) distr.  ((P  R)  (P  Q))  distr. ((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.
(DNF)
Chapter 1, Section 5
Discussion #10

13. CNF Conversion Example ( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
((P  Q)  R  (P  Q))  ((P  Q)  R  (P  Q)) impl.  (P  Q)  R  (P  Q) deM.  (P  Q)  R  (P  Q) deM.  (P  Q)  R  (P  Q) double neg.  ((P  R)  (Q  R))  (P  Q) distr.  ((P  R)  (P  Q))  distr. ((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.
(DNF)
CNF
Using the commutative and idempotent laws on the previous step and then the distributive law, we obtain this formula as the conjunctive normal form.
Chapter 1, Section 5
Discussion #10

14. CNF Conversion Example ( ..  ..  .. )  ( ..  ..  .. )  …  ( ..  .. )
(P  R)  (P  R  Q)
 (Q  R)
 (F  Q  R) - ident.
(P  R)  ((P  F)
 (Q  R)) - comm., distr.
 (P  R)  (F
 (Q  R)) - dominat.
(P  R)  (Q  R) - ident.
((P  Q)  R  (P  Q))  ((P  Q)  R  (P  Q)) impl.  (P  Q)  R  (P  Q) deM.  (P  Q)  R  (P  Q) deM.  (P  Q)  R  (P  Q) double neg.  ((P  R)  (Q  R))  (P  Q) distr.  ((P  R)  (P  Q))  distr. ((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.
(DNF)
Chapter 1, Section 5
Discussion #10

15. DNF Expression Generation
The only definition of  is the truth table F T  R Q P F T
(P  Q  R)
minterms
(P  Q  R)
(P  Q  R)
 (P  Q  R)  (P  Q  R)  (P  Q  R)
Chapter 1, Section 5
Discussion #10

16. CNF Expression Generation}
Find .
Find the DNF of .
Then, use De Morgan’s law to get the CNF of  (i.e. ()  )
Form a conjunction of max terms  F T  Q P
max terms T F
(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.
Chapter 1, Section 5
Discussion #10