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Forexample p3 py px py

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Unformatted text preview: denote

“x
>
0”
and
the
domain
be
 the
integers.
Then:
 P(‐3)


is
false.
 P(0)


is
false.
 P(3)

is
true.

   Often
the
domain
is
denoted
by
U.
So
in
this
example
U
is
 the
integers.
 Examples
of
Proposi*onal
 Func*ons
   Let
“x
+
y
=
z”
be
denoted
by

R(x,
y,
z)
and
U
(for
all
three
variables)
 be
the
integers.
Find
these
truth
values:

 R(2,-1,5)
 Solution:

F
 R(3,4,7)
 Solution:
T
 R(x,
3,
z)
 Solution:
Not
a
Proposition
   Now
let

“x
‐
y
=
z”
be
denoted
by
Q(x,
y,
z),
with
U
as
the
integers.
 Find
these
truth
values:
 Q(2,-1,3)
 
Solution:

T
 Q(3,4,7)
 
Solution:
F
 
Q(x,
3,
z)
 
Solution:

Not
a
Proposition
 Compound
Expressions
   Connectives
from
propositional
logic
carry
over
to
predicate
 logic.

   If
P(x)
denotes

“x
>
0,”
find
these
truth
values:
 P(3)
∨ P(-1) P(3)
∧ P(-1) P(3)
→ P(-1) P(3)
→ P(-1) Solution: T Solution: F Solution: F Solution: T
   Expressions
with
variables
are
not
propositions
and
therefore
 do
not
have
truth
values.

For
example,
 P(3)
∧ P(y) P(x)
→ P(y)   When
used
with
quantifiers
(to
be
introduced
next),
these
 expressions
(propositional
functions)
become
propositions.
 Quan*fiers
 Charles
Peirce
(1839‐1914)
   We
need
quantifiers
to
express
the
meaning
of
English
 words
including
all
and
some:
   “All
men
are
Mortal.”
   “Some
cats
do
not
have
fur.”
   The
two
most
important
quantifiers
are:
   Universal
Quantifier,
“For
all,”


symbol:
∀
   Existential
Quantifier,
...
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