Example
: “Every real number has an inverse” is
x
y(x + y = 0)
where the domains of x and y are the real numbers.
We can also think of nested propositional functions:
x
y(x + y = 0)
can be viewed as
x Q
(
x
)
where
Q
(
x
)
is
y P(x, y)
where
P(x, y)
is
(x + y = 0)

Thinking of Nested
Quantification
Nested Loops
To see if
x
yP (x,y)
is true, loop through the values of
x
:
At each step, loop through the values for
y
.
If for some pair of
x
and
y, P(x,y)
is false, then
x
yP(x,y)
is false and
both the outer and inner loop terminate.
x
y P(x,y)
is true
if the outer loop ends after stepping through each
x
.
To see if
x
yP(x,y)
is true, loop through the values of
x
:
At each step, loop through the values for
y.
The inner loop ends when a pair
x
and
y
is found such that
P
(
x
, y) is true.
If no
y
is found such that
P
(
x
, y) is true the outer loop terminates as
x
yP(x,y)
has
been shown to be false.
x
y P(x,y)
is true
if the outer loop ends after stepping through each
x
.
If the domains of the variables are infinite, then this
process can not actually be carried out.

Order of Quantifiers
Examples
:
1.
Let
P(x,y)
be the statement “
x + y = y +
x
.” Assume that
U
is the real
numbers. Then
x
yP(x,y)
and
y
xP(x,y)
have the same truth value.
2.
Let
Q(x,y)
be the statement “
x + y =
0
.” Assume that
U
is the real
numbers. Then
x
yP(x,y)
is true
,
but
y
xP(x,y)
is false.

Questions on Order of
Quantifiers
Example
1
: Let
U
be the real numbers,
Define
P(x,y) : x ∙ y
=
0
What is the truth value of the following:
1.
x
yP(x,y)
Answer:
False
2.
x
yP(x,y)
Answer:
True
3.
x
y P(x,y)
Answer:
True
4.
x
y P(x,y)
Answer:
True

Questions on Order of
Quantifiers
Example
2
: Let
U
be the real numbers,
Define
P(x,y)
:
x / y
=
1
What is the truth value of the following:
1.
x
yP(x,y)
Answer:
False
2.
x
yP(x,y)
Answer:
True
3.
x
y P(x,y)
Answer:
False
4.
x
y P(x,y)
Answer:
True

Quantifications of Two Variables
Statement
When True?
When False
P
(
x
,
y
) is true for every
pair
x
,
y
.
There is a pair
x, y
for
which
P
(
x,y
) is false.
For every
x
there is a
y
for which
P
(
x,y
) is true.
There is an x such that
P
(
x,y
) is false for every
y
.
There is an
x
for which
P
(
x,y
) is true for every
y
.
For every
x
there is a y
for which
P
(x,y) is false.
There is a pair
x, y
for
which
P
(
x,y
) is true.
P
(x,y) is false for every
pair
x,y

Translating Nested Quantifiers into
English
Example
1
: Translate the statement
x
(C(x )∨
y (C(y ) ∧ F(x, y)))
where C(x) is “
x
has a computer,” and
F
(
x
,
y
) is “
x
and
y
are friends,” and the
domain for both
x
and
y
consists of all
students in your school.
Solution
: Every student in your school has
a computer or has a friend who has a
computer.

Translating Mathematical
Statements into Predicate Logic
Example
:
Translate “The sum of two positive
integers is always positive” into a logical
expression.