Math 135: Lecture 8: Nested Quantifiers
Definition 8.1.
Let
S
and
T
be two sets. A function
f
:
S
→
T
is
(or
) if
and only if for every
y
∈
T
there exists an
x
∈
S
so that
f
(
x
) =
y
.
Often
S
and
T
are equal to
R
or are subsets of
R
.
Nested Quantifiers
1. Process quantifiers from left to right.
2. Use existing construction and choose techniques as you proceed from left to right.
Example 8.2
(Order of Quantifiers is Important)
.
∀
x
∃
y, y > x
“Given any integer
x
, there exists a larger integer
y
.”
(A
statement.)
∃
y
∀
x, y > x
“There exists an integer
y
which is larger than all integers.”
(A
statement.)
Onto
To say that the function
f
:
S
→
T
onto means:
For every
y
∈
T
there exists
x
∈
S
so that
f
(
x
) =
y
.
Object:
Universe of discourse:
Certain property:
Something happens:
The “something that happens” contains a nested existential quantifier with the following
components.
Object:
Universe of discourse:
Certain property:
Something happens:
Example 8.3.
The function
g
:
R
→
R
defined by
g
(
x
) =
x
2
is NOT onto.
Proof:
1. Let
y
=

2
∈
R
.
Is there an
x
so that
g
(
x
) =
y
?
2. Since
g
(
x
) =
x
2
there is no
x
∈
R
such that
g
(
x
) =

2.
3. Thus
g
(
x
) is not onto.
1
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Math 135: Lecture 8: Nested Quantifiers
Proposition 8.4.
The function
f
:
R
→
R
defined by
f
(
x
) =
x
3
is onto.
Proof of Proposition ??:
1. Let
y
∈
R
.
Can we find
x
so that
f
(
x
) =
y
?
2. Consider
x
=
3
√
y
.
3. Then
f
(
x
) =
f
(
3
√
y
) = (
3
√
y
)
3
=
y
as needed.
Analysis of Proof
The definition of
onto
uses a nested quantifier.
Hypothesis:
Conclusion:
Core Proof Technique:
Nested quantifiers.
Preliminaries:
Onto
as it applies in this situation.
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 Fall '08
 ANDREWCHILDS
 Math, Sets, Semantics, Philosophy of mathematics, certain property

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