Transcript CS 236 – Discrete Mathematics

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
Name
Excluded middle law
Contradiction law
Identity laws
Domination laws
Idempotent laws
Double-negation law
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
Name
Commutative laws
Associative laws
Distributive laws
De Morgan’s laws
Absorption laws
Laws of  and 
Law
P  Q  P  Q
Name
Implication Law
P Q  PQQP
Equivalence Law
P  Q  Q  P
Contrapositive Law
P  Q  P  Q  F
Contradiction Law
Main Rules of Inference
A, B |= A  B
Law of combination
A  B |= B
Law of simplification
A  B |= A
Variant of law of simplification
A |= A  B
Law of addition
B |= A  B
Variant of law of addition
A, AB |= B
Modus ponens
B, AB |= A
Modus tollens
AB, BC |= AC
Hypothetical syllogism
A  B, A |= B
Disjunctive syllogism
A  B, B |= A
Variant of disjunctive syllogism
AB, AB |= B
Law of cases
AB |= AB
Equivalence elimination
AB |= BA
Variant of equivalence elimination
AB, BA |= AB
Equivalence introduction
A, A |= B
Inconsistency law