Download Quantifiers Lecture 7 Robb T. Koether Wed, Feb 3, 2016

Quantifiers
Lecture 7
Robb T. Koether
Hampden-Sydney College
Wed, Feb 3, 2016
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
1 / 28
Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Free Variables
Definition (Free Variable)
A variable x in a sentence is free if it is simply a placeholder for a
value, but by itself has no value.
For example, in the sentence “x is divisible by 4”, x is a free
variable.
If a sentence contains a free variable, then the sentence cannot
be either true or false.
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Bound Variables
Definition (Bound Variable)
A variable x in a sentence is bound if the sentence binds it to a specific
set of values.
For example, in the sentence “There is an integer x that is
divisible by 4”, x is a bound variable.
If all the variables in a sentence are bound, then the sentence is
either true or false.
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Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
We use two quantifiers to bind variables.
The universal quantifier is the phrase
“for all . . . in the set . . . ”
The existential quantifier is the phrase
“there exists . . . in the set . . . such that . . . ”
We use the symbols ∀ and ∃ to represent the universal and
existential quantifiers, respectively.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
Example (Quantifiers)
Write each sentence using the qualifier symbols and then identify
the sentence as (1) a true statement, (2) a false statement, or (3)
neither.
1
For all real numbers x, x 2 is positive.
Equivalently, For all x in the set R, x 2 is positive.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
Example (Quantifiers)
Write each sentence using the qualifier symbols and then identify
the sentence as (1) a true statement, (2) a false statement, or (3)
neither.
1
For all real numbers x, x 2 is positive.
Equivalently, For all x in the set R, x 2 is positive.
2
For all rational numbers q, q 2 is rational.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
Example (Quantifiers)
Write each sentence using the qualifier symbols and then identify
the sentence as (1) a true statement, (2) a false statement, or (3)
neither.
1
For all real numbers x, x 2 is positive.
Equivalently, For all x in the set R, x 2 is positive.
2
3
For all rational numbers q, q 2 is rational.
There exists a real number x such that x 2 + x − 1 = 0.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
Example (Quantifiers)
Write each sentence using the qualifier symbols and then identify
the sentence as (1) a true statement, (2) a false statement, or (3)
neither.
1
For all real numbers x, x 2 is positive.
Equivalently, For all x in the set R, x 2 is positive.
2
3
4
For all rational numbers q, q 2 is rational.
There exists a real number x such that x 2 + x − 1 = 0.
There exists a real number x such that x 5 + x − 1 = 0.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
8 / 28
The Quantifiers
Example (Quantifiers)
Write each sentence using the qualifier symbols and then identify
the sentence as (1) a true statement, (2) a false statement, or (3)
neither.
1
For all real numbers x, x 2 is positive.
Equivalently, For all x in the set R, x 2 is positive.
2
3
4
5
For all rational numbers q, q 2 is rational.
There exists a real number x such that x 2 + x − 1 = 0.
There exists a real number x such that x 5 + x − 1 = 0.
For every real number x, there exists a real number y such that
x + y = 1.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
Example (Quantifiers)
Write each sentence using the qualifier symbols and then identify
the sentence as (1) a true statement, (2) a false statement, or (3)
neither.
1
For all real numbers x, x 2 is positive.
Equivalently, For all x in the set R, x 2 is positive.
2
3
4
5
6
For all rational numbers q, q 2 is rational.
There exists a real number x such that x 2 + x − 1 = 0.
There exists a real number x such that x 5 + x − 1 = 0.
For every real number x, there exists a real number y such that
x + y = 1.
There exists real number x such that for every a real number y ,
x + y = 1.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
Example (Quantifiers)
Write each sentence using the qualifier symbols and then identify
the sentence as (1) a true statement, (2) a false statement, or (3)
neither.
1
For all real numbers x, x 2 is positive.
Equivalently, For all x in the set R, x 2 is positive.
2
3
4
5
6
7
For all rational numbers q, q 2 is rational.
There exists a real number x such that x 2 + x − 1 = 0.
There exists a real number x such that x 5 + x − 1 = 0.
For every real number x, there exists a real number y such that
x + y = 1.
There exists real number x such that for every a real number y ,
x + y = 1.
There exists an integer a such that for every integer b there exists
an integer c such that a + bc 6= 0.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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The Quantifiers
Example (Quantifiers)
Consider “There exists an integer a such that for every integer b
there exists an integer c such that a + bc 6= 0.”
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Quantifiers
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The Quantifiers
Example (Quantifiers)
Consider “There exists an integer a such that for every integer b
there exists an integer c such that a + bc 6= 0.”
(∃a ∈ Z)(∀b ∈ Z)(∃c ∈ Z)(a + bc 6= 0).
Robb T. Koether (Hampden-Sydney College)
Quantifiers
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The Quantifiers
Example (Quantifiers)
Consider “There exists an integer a such that for every integer b
there exists an integer c such that a + bc 6= 0.”
(∃a ∈ Z)(∀b ∈ Z)(∃c ∈ Z)(a + bc 6= 0).
You choose a, hoping to make a + bc 6= 0.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
9 / 28
The Quantifiers
Example (Quantifiers)
Consider “There exists an integer a such that for every integer b
there exists an integer c such that a + bc 6= 0.”
(∃a ∈ Z)(∀b ∈ Z)(∃c ∈ Z)(a + bc 6= 0).
You choose a, hoping to make a + bc 6= 0.
Your worst enemy chooses b, hoping to force a + bc = 0.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
9 / 28
The Quantifiers
Example (Quantifiers)
Consider “There exists an integer a such that for every integer b
there exists an integer c such that a + bc 6= 0.”
(∃a ∈ Z)(∀b ∈ Z)(∃c ∈ Z)(a + bc 6= 0).
You choose a, hoping to make a + bc 6= 0.
Your worst enemy chooses b, hoping to force a + bc = 0.
You choose c, hoping to make a + bc 6= 0.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
9 / 28
The Quantifiers
Example (Quantifiers)
Consider “There exists an integer a such that for every integer b
there exists an integer c such that a + bc 6= 0.”
(∃a ∈ Z)(∀b ∈ Z)(∃c ∈ Z)(a + bc 6= 0).
You choose a, hoping to make a + bc 6= 0.
Your worst enemy chooses b, hoping to force a + bc = 0.
You choose c, hoping to make a + bc 6= 0.
Can you do it?
Can your worst enemy prevent you from doing it?
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
9 / 28
The Order of Quantifiers
Note that the order of the quantifiers ∀ and ∃ cannot be changed
without changing the meaning of the statement.
However, the order of consecutive occurrences of ∀ or
consecutive occurrences of ∃ can be reversed without changing
the meaning of the statement.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
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The Order of Quantifiers
Example
The statements
(∀a ∈ Z)(∀b ∈ Z)(a + b ∈ Z)
and
(∀b ∈ Z)(∀a ∈ Z)(a + b ∈ Z)
are logically equivalent.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
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The Order of Quantifiers
Example
The statements
(∃a ∈ Z)(∃b ∈ Z)(ab = 6)
and
(∃b ∈ Z)(∃a ∈ Z)(ab = 6)
are logically equivalent.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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If a proposition is not true for every x in a set, then it must be false
for at least one x in that set.
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If a proposition is not true for every x in a set, then it must be false
for at least one x in that set.
Therefore,
¬ (∀x ∈ S) (P(x))
is equivalent to
(∃x ∈ S) (¬P(x)) .
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Similarly, if a proposition is not true for even one x in a set, then it
must be false for all x in that set.
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Similarly, if a proposition is not true for even one x in a set, then it
must be false for all x in that set.
Therefore,
¬ (∃x ∈ S) (P(x))
is equivalent to
(∀x ∈ S) (¬P(x)) .
Robb T. Koether (Hampden-Sydney College)
Quantifiers
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Write the negations of the following propositions and then
determine which is true, the proposition or its negation.
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Quantifiers
Wed, Feb 3, 2016
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Write the negations of the following propositions and then
determine which is true, the proposition or its negation.
1
(∀a ∈ Z) (∀b ∈ Z) ((ab is even) ⇒ ((a is even) ∨ (b is even))).
Robb T. Koether (Hampden-Sydney College)
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Write the negations of the following propositions and then
determine which is true, the proposition or its negation.
1
2
(∀a ∈ Z) (∀b ∈ Z) ((ab is even) ⇒ ((a is even) ∨ (b is even))).
(∀a ∈ Z) (∀b ∈ Z) ((4 | ab) ⇒ (4 | a) ∨ (4 | b))).
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Write the negations of the following propositions and then
determine which is true, the proposition or its negation.
1
2
3
(∀a ∈ Z) (∀b ∈ Z) ((ab is even) ⇒ ((a is even) ∨ (b is even))).
(∀a ∈ Z) (∀b ∈ Z) ((4 | ab) ⇒ (4| a) ∨ (4 | b))).
(∃a ∈ Z) (∃b ∈ Z) a2 + b2 = 99 .
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Write the negations of the following propositions and then
determine which is true, the proposition or its negation.
1
2
3
4
(∀a ∈ Z) (∀b
(∀a ∈ Z) (∀b
(∃a ∈ Z) (∃b
(∃a ∈ Z) (∃b
∈ Z) ((ab is even) ⇒ ((a is even) ∨ (b is even))).
∈ Z) ((4 | ab) ⇒ (4| a) ∨ (4 | b))).
∈ Z) a2 + b2 = 99 .
∈ Z) a2 − b2 = 99 .
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Write the negations of the following propositions and then
determine which is true, the proposition or its negation.
1
2
3
4
5
(∀a ∈ Z) (∀b
(∀a ∈ Z) (∀b
(∃a ∈ Z) (∃b
(∃a ∈ Z) (∃b
(∀a ∈ Z) (∃b
∈ Z) ((ab is even) ⇒ ((a is even) ∨ (b is even))).
∈ Z) ((4 | ab) ⇒ (4| a) ∨ (4 | b))).
∈ Z) a2 + b2 = 99 .
∈ Z) a2 − b2 = 99 .
∈ Z) (∃c ∈ Z) (a = 2b + 3c).
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Proving a Universal Statement
To prove directly a statement of the form “(∀x ∈ S) (P(x)),” begin
with “Let x ∈ S.”
Do not assume anything about x except that it is a member of the
set S.
Then go on to argue that P(x) is true.
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Proving an Existential Statement
To prove directly a statement of the form “(∃x ∈ S) (P(x)),” you
can proceed in one of two ways.
Produce a specific value of x for which P(x) is true.
Argue indirectly that such an x must exist.
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Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Style
The symbols ∀ and ∃ are universally recognized.
Nevertheless, we should avoid using them when writing ordinary
sentences.
Reserve their use for situations where we are dealing explicitly
with logical forms.
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Good Style
Theorem (Good Style)
The sum of two even integers is even.
Proof.
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Good Style
Theorem (Good Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
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Good Style
Theorem (Good Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
Then there exists integers s and t such that
n = 2s,
m = 2t.
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Good Style
Theorem (Good Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
Then there exists integers s and t such that
n = 2s,
m = 2t.
It follows that n + m = 2s + 2t = 2(s + t).
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Good Style
Theorem (Good Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
Then there exists integers s and t such that
n = 2s,
m = 2t.
It follows that n + m = 2s + 2t = 2(s + t).
Therefore, n + m is even.
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Poor Style
Theorem (Poor Style)
The sum of two even integers is even.
Proof.
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Poor Style
Theorem (Poor Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
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Poor Style
Theorem (Poor Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
Then (∃s ∈ Z)(n = 2s) and (∃t ∈ Z)(m = 2t)
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Poor Style
Theorem (Poor Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
Then (∃s ∈ Z)(n = 2s) and (∃t ∈ Z)(m = 2t)
It follows that n + m = 2s + 2t = 2(s + t).
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Poor Style
Theorem (Poor Style)
The sum of two even integers is even.
Proof.
Let n and m be even integers.
Then (∃s ∈ Z)(n = 2s) and (∃t ∈ Z)(m = 2t)
It follows that n + m = 2s + 2t = 2(s + t).
Therefore, n + m is even.
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
23 / 28
Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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A Little More LATEX
The quantifiers (in math mode)
For every – \forall will produce ∀.
There exists – \exists will produce ∃.
Set membership (in math mode)
Is an element of – \in will produce ∈.
Is not an element of – \notin will produce ∈.
/
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Example
Example (A Little More LATEX)
The LATEX expression
(\forall n\in\mathbb{Z})(\exists
m\in\mathbb{Z})(n+m=0)
produces
(∀n ∈ Z)(∃m ∈ Z)(n + m = 0)
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Outline
1
Free and Bound Variables
2
Quantifiers
3
Negation of Quantifiers
4
Writing Proofs
5
Style
6
A Little More LATEX
7
Assignment
Robb T. Koether (Hampden-Sydney College)
Quantifiers
Wed, Feb 3, 2016
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Assignment
Homework
Presentations this week:
Pr 1.111, 1.112, 1.113, 1.114, 1.116, 1.117, 1.119.
Write up exactly 2 of the 3 problems Ex 1.92, Pr 1.115, and Pr
1.118 to turn in on Wednesday, February 10.
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