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Introduction
Rules of Inference
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Rules of Inference
52
Previous Lecture
Logically equivalent statements
Statements
Φ
and
Ψ
are equivalent iff
Φ↔Ψ
is a tautology
Main logic equivalences
s
double negation
s
DeMorgan’s laws
s
commutative, associative, and distributive laws
s
idempotent, identity, and domination laws
s
the law of contradiction and the law of excluded middle
s
absorption laws
Discrete Mathematics – Rules of Inference
43
Expressing Connectives
Some connectives can be expressed through others
s
p
⊕
q
⇔ ¬
(p
↔
q)
s
p
↔
q
⇔
(p
→
q)
∧
(q
→
p)
s
p
→
q
⇔ ¬
p
∨
q
Theorem
Every compound statement is logically equivalent to a
statement that uses only conjunction, disjunction, and negation
Discrete Mathematics – Rules of Inference
44
Example
Simplify the statement
(p
∨
q)
↔
(p
→
q)
Discrete Mathematics – Rules of Inference
55
First Law of Substitution
Suppose that the compound statement
Φ
is a tautology.
If
p
is a
primitive statement that appears in
Φ
and we replace each
occurrence of
p
by the same statement
q (not necessarily
primitive), then the resulting compound statement
Ψ
is also a
tautology.
Let
Φ
= (p
→
q)
∨
(q
→
p),
and we substitute
p
by
p
∨
(s
⊕
r)
Therefore
((p
∨
(s
⊕
r))
→
q)
∨
(q
→
(p
∨
(s
⊕
r))
is a tautology
Discrete Mathematics – Rules of Inference
56
Second Law of Substitution
Let
Φ
be a compound statement,
p
an arbitrary (not necessarily
primitive!) statement that appears in
Φ
,
and let
q
be a statement
such that
p
⇔
q.
If we replace one or more occurrences of
p
by
q,
then for the resulting compound statement
Ψ
we have
Φ ⇔ Ψ
.
Let
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 Spring '08
 PEARCE

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