Introduction
Logic Equivalence
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Logic Equivalence
92
Previous Lecture
Free and bound variables
Multiple quantifiers and logic connectives
Definitions, rules, and theorems
Discrete Mathematics – Logic Equivalence
93
Rules
Predicates and quantifiers are (implicitly) present in rules and laws
``Everyone having income more that $20000 must file a tax report’’
P(x)  ``x
has income more than $20000’’
Q(x)  ``x
must file a tax report’’
2200
x (P(x)
→
Q(x))
Discrete Mathematics – Logic Equivalence
94
Theorems
Every theorem involves predicates and quantifiers
``For every statement there is an equivalent CNF’’
C(x)  ``x
is a CNF’’
2200
x
5
y (C(y)
∧
(x
⇔
y))
``A parallelogram is a rectangle if all its angles are equal’’
R(x)  ``parallelogram
x
is a rectangle’’
A(x)  ``all angles of
x
are equal’’
2200
x (A(x)
→
R(x))
Discrete Mathematics – Logic Equivalence
95
Universe and Interpretation
A logic statement is meaningless
2200
x P(x)
It only makes sense if we
specify a universe and a
particular meaning of the
predicate
universe: animals
P(x):
x
has horns
universe: cars
P(x):
x
is red
universe: numbers
P(x):
x
is even
Interpretation
Discrete Mathematics – Logic Equivalence
96
Logical Equivalence of Predicates
Recall that two compound statements
Φ
and
Ψ
are logically
equivalent
(
Φ ⇔ Ψ
)
if and only if
Φ ↔ Ψ
is a tautology.
For predicates:
Two predicates
P(x)
and
Q(x)
are
logically equivalent
in a
given universe if and only if, for any value
a
from the universe
statements
P(a)
and
Q(a)
are equivalent
if and only if the statement
2200
x (P(x)
↔
Q(x))
is true in the given
universe
``A parallelogram is a rectangle if and only if all its angles are equal’’
P(x)  ``x is a rectangle’’
Q(x)  ``all angles of
x are equal’’
P(x)
⇔
Q(x)
in the universe of parallelograms
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97
Logical Equivalence of Quantified Statements
Two quantified statements are said to be
logically equivalent
if
they are equivalent for any given universe.
Consider statements
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 Spring '08
 PEARCE
 Logic, Parallelograms, equivalences, Logic Equivalence

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