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Otherwiseitisunsatisable examplenotvalidbutsatisable

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Unformatted text preview: g
“x
is
 mortal.”

Specify
the

domain
as
all
people.
   The
two
premises
are:
   The
conclusion
is:
   Later
we
will
show
how
to
prove
that
the
conclusion
 follows
from
the
premises.
 Equivalences
in
Predicate
Logic
   Statements
involving
predicates
and
quantifiers
are
 logically
equivalent
if
and
only
if
they
have
the
same
 truth
value

   for
every
predicate
substituted
into
these
statements
 and

   for
every
domain
of
discourse
used
for
the
variables
in
 the
expressions.

   The
notation
S
≡T indicates that S and T are logically equivalent.   Example: ∀x ¬¬S(x) ≡ ∀x S(x)
 Thinking
about
Quan*fiers
as
 Conjunc*ons
and
Disjunc*ons
   If
the
domain
is
finite,
a
universally
quantified
proposition
is
 equivalent
to
a
conjunction
of
propositions
without
quantifiers
 and
an
existentially
quantified
proposition
is
equivalent
to

a
 disjunction
of
propositions
without
quantifiers.

   If
U
consists
of
the
integers
1,2,
and
3:
   Even
if
the
domains
are
infinite,
you
can
still
think
of
the
 quantifiers
in
this
fashion,
but
the
equivalent
expressions
 without
quantifiers
will
be
infinitely
long.
 Nega*ng
Quan*fied
Expressions
   Consider
∀x J(x)
 “Every
student
in
your
class
has
taken
a
course
in
Java.”
 
Here
J(x)

is
“x
has
taken
a
course
in
calculus”
and

 
the
domain
is
students
in
your
class.

   Negating
the
original
statement
gives
“It
is
not
the
 case
that
every
student
in
your
class
has
taken
Java.”
 This
implies
that
“There
is...
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This document was uploaded on 03/06/2014 for the course MATH 320 at CSU Northridge.

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