lecture notes10

Ifatsomesteppxistruethenxpxistrueandtheloop terminates

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Unformatted text preview: There
exists,”

symbol:
∃
   We
write

as
in
∀x
P(x)
and
∃x
P(x).
   ∀x
P(x)
asserts
P(x)
is
true
for
every
x
in
the
domain.
   ∃x
P(x)
asserts
P(x)
is
true
for
some
x
in
the
domain.
   The
quantifiers
are
said
to
bind
the
variable
x
in
these
 expressions.

 Universal
Quan*fier
   ∀x
P(x)

is
read
as
“For
all
x,
P(x)”
or
“For
every
x,
P(x)”
 Examples:
 1)  2)  3)  
If
P(x)
denotes

“x
>
0” and U is the integers, then ∀x
P(x)
 is
false.
 If
P(x)
denotes

“x
>
0” and U is the positive integers, then ∀x
P(x)
is
true.
 If
P(x)
denotes

“x
is
even” and U is the integers, then ∀
x
 P(x)
is
false.
 Existen*al
Quan*fier
   ∃x
P(x)
is
read
as
“For
some
x,
P(x)”,

or
as
“There
is
 an
x
such
that
P(x),”

or
“For
at
least
one
x,
P(x).”

 Examples:
 1.  2.  3.  
If
P(x)
denotes

“x
>
0” and U is the integers, then ∃x
P(x)
 is
true.
It
is
also
true
if
U
is
the
positive
integers.
 If
P(x)
denotes

“x
<
0” and U is the positive integers, then ∃x
P(x)
is
false.
 If
P(x)
denotes

“x
is
even” and U is the integers, then ∃x
P(x)
is
true.
 Uniqueness
Quan*fier
(op#onal)
   ∃!x
P(x)
means
that
P(x)
is
true
for
one
and
only
one
x in
the
 universe
of
discourse.
   This
is
commonly
expressed
in
English
in
the
following
 equivalent
ways:
   “There
is
a
unique
x
such
that
P(x).”

   “There
is
one
and
only
one
x
such
that
P(x)”
   Examples:
 1.  If
P(x)
denotes

“x
+
1
=
0” and U is the integers, then ∃!x
P(x)
is...
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