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§ 1.6 Nested Quantifiers
The Beginning
When we talk of nested quantifiers, we mean statements like
∀ x ∃ y (x < y)
This is the same thing as ∀ x Q(x), where Q(x) is ∃ y P(x, y), whereP(x, y) is x > y. This is where the term ‘nested’ comes from.
To understand the statements involving multiple quantifiers, we needto make sure we understand what the quantifiers and predicates thatwe see mean.
The Beginning
When we talk of nested quantifiers, we mean statements like
∀ x ∃ y (x < y)
This is the same thing as ∀ x Q(x), where Q(x) is ∃ y P(x, y), whereP(x, y) is x > y. This is where the term ‘nested’ comes from.
To understand the statements involving multiple quantifiers, we needto make sure we understand what the quantifiers and predicates thatwe see mean.
The Beginning
When we talk of nested quantifiers, we mean statements like
∀ x ∃ y (x < y)
This is the same thing as ∀ x Q(x), where Q(x) is ∃ y P(x, y), whereP(x, y) is x > y. This is where the term ‘nested’ comes from.
To understand the statements involving multiple quantifiers, we needto make sure we understand what the quantifiers and predicates thatwe see mean.
Examples
ExampleAssume that the domain of discourse for the variables x and y consistof all real numbers (R×R).
The statement∀ x ∀ y (x + y = y + x)
says that x + y = y + x for all real numbers x and y. This is thecommutative law for the addition of real numbers.
We could have written this in terms of the predicate P(x, y) whereP(x, y) is x + y = y + x.
Likewise, the statement
∀ x ∃ y (x + y = 0)
says that for every real number x there is a real number y such thatx + y = 0. This is the additive inverse property.
Examples
ExampleAssume that the domain of discourse for the variables x and y consistof all real numbers (R×R).The statement
∀ x ∀ y (x + y = y + x)
says that x + y = y + x for all real numbers x and y. This is thecommutative law for the addition of real numbers.
We could have written this in terms of the predicate P(x, y) whereP(x, y) is x + y = y + x.
Likewise, the statement
∀ x ∃ y (x + y = 0)
says that for every real number x there is a real number y such thatx + y = 0. This is the additive inverse property.
Examples
ExampleAssume that the domain of discourse for the variables x and y consistof all real numbers (R×R).The statement
∀ x ∀ y (x + y = y + x)
says that x + y = y + x for all real numbers x and y. This is thecommutative law for the addition of real numbers.
We could have written this in terms of the predicate P(x, y) whereP(x, y) is x + y = y + x.
Likewise, the statement
∀ x ∃ y (x + y = 0)
says that for every real number x there is a real number y such thatx + y = 0. This is the additive inverse property.
Examples
ExampleAssume that the domain of discourse for the variables x and y consistof all real numbers (R×R).The statement
∀ x ∀ y (x + y = y + x)
says that x + y = y + x for all real numbers x and y. This is thecommutative law for the addition of real numbers.
We could have written this in terms of the predicate P(x, y) whereP(x, y) is x + y = y + x.
Likewise, the statement
∀ x ∃ y (x + y = 0)
says that for every real number x there is a real number y such thatx + y = 0. This is the additive inverse property.
Examples
ExampleAssume that the domain of discourse for the variables x and y consistof all real numbers (R×R).The statement
∀ x ∀ y (x + y = y + x)
says that x + y = y + x for all real numbers x and y. This is thecommutative law for the addition of real numbers.
We could have written this in terms of the predicate P(x, y) whereP(x, y) is x + y = y + x.
Likewise, the statement
∀ x ∃ y (x + y = 0)
says that for every real number x there is a real number y such thatx + y = 0. This is the additive inverse property.
Examples (cont.)
ExampleFinally, the statement
∀ x ∀ y ∀ z (x + (y + z) = (x + y) + z)
is the associative law for addition of real numbers.
ExampleAssume the domain of discourse isR×R. Translate the followinginto English:
∀ x ∀ y ((x > 0) ∧ (y < 0)→ (xy < 0))
The statement says that for every real number x and every real numbery, if x > 0 and y < 0, then xy < 0. That is, the product of a positivereal number and a negative real number is a negative real number.
Examples (cont.)
ExampleFinally, the statement
∀ x ∀ y ∀ z (x + (y + z) = (x + y) + z)
is the associative law for addition of real numbers.
ExampleAssume the domain of discourse isR×R. Translate the followinginto English:
∀ x ∀ y ((x > 0) ∧ (y < 0)→ (xy < 0))
The statement says that for every real number x and every real numbery, if x > 0 and y < 0, then xy < 0. That is, the product of a positivereal number and a negative real number is a negative real number.
Examples (cont.)
ExampleFinally, the statement
∀ x ∀ y ∀ z (x + (y + z) = (x + y) + z)
is the associative law for addition of real numbers.
ExampleAssume the domain of discourse isR×R. Translate the followinginto English:
∀ x ∀ y ((x > 0) ∧ (y < 0)→ (xy < 0))
The statement says that for every real number x and every real numbery, if x > 0 and y < 0, then xy < 0. That is, the product of a positivereal number and a negative real number is a negative real number.
Does Order Matter?
Example
Assume the domain of discourse isR×R. Let P(x, y) be thestatement ‘x + y = y + x’. What are the truth values for thequantification ∀ x ∀ y P(x, y) and ∀ y ∀ x P(x, y)?
Because P(x, y) is true for all real numbers x and all real numbers y,the statement ∀ x ∀ y P(x, y) is true.
Notice that ∀ y ∀ x P(x, y) states that for all real numbers y and all realnumbers x, x + y = y + x. So both statements have the same meaning.
Does Order Matter?
Example
Assume the domain of discourse isR×R. Let P(x, y) be thestatement ‘x + y = y + x’. What are the truth values for thequantification ∀ x ∀ y P(x, y) and ∀ y ∀ x P(x, y)?
Because P(x, y) is true for all real numbers x and all real numbers y,the statement ∀ x ∀ y P(x, y) is true.
Notice that ∀ y ∀ x P(x, y) states that for all real numbers y and all realnumbers x, x + y = y + x. So both statements have the same meaning.
Does Order Matter?
Example
Assume the domain of discourse isR×R. Let P(x, y) be thestatement ‘x + y = y + x’. What are the truth values for thequantification ∀ x ∀ y P(x, y) and ∀ y ∀ x P(x, y)?
Because P(x, y) is true for all real numbers x and all real numbers y,the statement ∀ x ∀ y P(x, y) is true.
Notice that ∀ y ∀ x P(x, y) states that for all real numbers y and all realnumbers x, x + y = y + x. So both statements have the same meaning.
Does Order Matter
Example
Assume the domain of discourse isR×R. Let Q(x, y) be thestatement x + y = 0. What are the truth values for the quantification∀ x ∃ y Q(x, y) and for ∃ y ∀ x Q(x, y)?
We have already seen ∀ x ∃ y (Q(x, y)) and we know this is a truestatement.
What does the statement ∃ y ∀ x (Q(x, y)) say?
The statement says ‘There is a number y such that for every realnumber x, Q(x, y).
No matter what value of y is chosen, there is only one value x thatmakes x + y = 0 true, so the statement ∃ y ∀ x Q(x, y) is false.
Does Order Matter
Example
Assume the domain of discourse isR×R. Let Q(x, y) be thestatement x + y = 0. What are the truth values for the quantification∀ x ∃ y Q(x, y) and for ∃ y ∀ x Q(x, y)?
We have already seen ∀ x ∃ y (Q(x, y)) and we know this is a truestatement.
What does the statement ∃ y ∀ x (Q(x, y)) say?
The statement says ‘There is a number y such that for every realnumber x, Q(x, y).
No matter what value of y is chosen, there is only one value x thatmakes x + y = 0 true, so the statement ∃ y ∀ x Q(x, y) is false.
Does Order Matter
Example
Assume the domain of discourse isR×R. Let Q(x, y) be thestatement x + y = 0. What are the truth values for the quantification∀ x ∃ y Q(x, y) and for ∃ y ∀ x Q(x, y)?
We have already seen ∀ x ∃ y (Q(x, y)) and we know this is a truestatement.
What does the statement ∃ y ∀ x (Q(x, y)) say?
The statement says ‘There is a number y such that for every realnumber x, Q(x, y).
No matter what value of y is chosen, there is only one value x thatmakes x + y = 0 true, so the statement ∃ y ∀ x Q(x, y) is false.
Does Order Matter
Example
Assume the domain of discourse isR×R. Let Q(x, y) be thestatement x + y = 0. What are the truth values for the quantification∀ x ∃ y Q(x, y) and for ∃ y ∀ x Q(x, y)?
We have already seen ∀ x ∃ y (Q(x, y)) and we know this is a truestatement.
What does the statement ∃ y ∀ x (Q(x, y)) say?
The statement says ‘There is a number y such that for every realnumber x, Q(x, y).
No matter what value of y is chosen, there is only one value x thatmakes x + y = 0 true, so the statement ∃ y ∀ x Q(x, y) is false.
Does Order Matter
Example
Assume the domain of discourse isR×R. Let Q(x, y) be thestatement x + y = 0. What are the truth values for the quantification∀ x ∃ y Q(x, y) and for ∃ y ∀ x Q(x, y)?
We have already seen ∀ x ∃ y (Q(x, y)) and we know this is a truestatement.
What does the statement ∃ y ∀ x (Q(x, y)) say?
The statement says ‘There is a number y such that for every realnumber x, Q(x, y).
No matter what value of y is chosen, there is only one value x thatmakes x + y = 0 true, so the statement ∃ y ∀ x Q(x, y) is false.
Does Order Matter
Conclusion? Do you think order matters with nested quantifiers?
What we have is
∀ x ∀ y P(x, y) ≡ ∀ y ∀ x P(x, y)
∃ y ∀ x Q(x, y) 6≡ ∀ x ∃ y Q(x, y)
This makes it seem that order matters when we use differentquantifiers.
Does Order Matter
Conclusion? Do you think order matters with nested quantifiers?
What we have is
∀ x ∀ y P(x, y) ≡ ∀ y ∀ x P(x, y)
∃ y ∀ x Q(x, y) 6≡ ∀ x ∃ y Q(x, y)
This makes it seem that order matters when we use differentquantifiers.
Does Order Matter
Conclusion? Do you think order matters with nested quantifiers?
What we have is
∀ x ∀ y P(x, y) ≡ ∀ y ∀ x P(x, y)
∃ y ∀ x Q(x, y) 6≡ ∀ x ∃ y Q(x, y)
This makes it seem that order matters when we use differentquantifiers.
Quantification of Two Variables
Quantifications of Two VariablesStatement When True? When False?∀ x ∀ y P(x, y) P(x, y) is true for There is a pair for∀ y ∀ x P(x, y) all pairs x, y which P(x, y) is false
∀ x ∃ y P(x, y) For every x there is a y There is an x such thatfor which P(x, y) is true P(x, y) is false for every y
∃ x ∀ y P(x, y) There is an x for which For every x there is a y forP(x, y) is true for every y which P(x, y) is false
∃ x ∃ y P(x, y) There is a pair x, y for P(x, y) is false for every∃ y ∃ x P(x, y) which P(x, y) is true pair x, y
Quantification of Two Variables
Quantifications of Two VariablesStatement When True? When False?∀ x ∀ y P(x, y) P(x, y) is true for There is a pair for∀ y ∀ x P(x, y) all pairs x, y which P(x, y) is false∀ x ∃ y P(x, y) For every x there is a y There is an x such that
for which P(x, y) is true P(x, y) is false for every y
∃ x ∀ y P(x, y) There is an x for which For every x there is a y forP(x, y) is true for every y which P(x, y) is false
∃ x ∃ y P(x, y) There is a pair x, y for P(x, y) is false for every∃ y ∃ x P(x, y) which P(x, y) is true pair x, y
Quantification of Two Variables
Quantifications of Two VariablesStatement When True? When False?∀ x ∀ y P(x, y) P(x, y) is true for There is a pair for∀ y ∀ x P(x, y) all pairs x, y which P(x, y) is false∀ x ∃ y P(x, y) For every x there is a y There is an x such that
for which P(x, y) is true P(x, y) is false for every y∃ x ∀ y P(x, y) There is an x for which For every x there is a y for
P(x, y) is true for every y which P(x, y) is false
∃ x ∃ y P(x, y) There is a pair x, y for P(x, y) is false for every∃ y ∃ x P(x, y) which P(x, y) is true pair x, y
Quantification of Two Variables
Quantifications of Two VariablesStatement When True? When False?∀ x ∀ y P(x, y) P(x, y) is true for There is a pair for∀ y ∀ x P(x, y) all pairs x, y which P(x, y) is false∀ x ∃ y P(x, y) For every x there is a y There is an x such that
for which P(x, y) is true P(x, y) is false for every y∃ x ∀ y P(x, y) There is an x for which For every x there is a y for
P(x, y) is true for every y which P(x, y) is false∃ x ∃ y P(x, y) There is a pair x, y for P(x, y) is false for every∃ y ∃ x P(x, y) which P(x, y) is true pair x, y
An Example
Example
Let Q(x, y, z) be the statement x + y = z. What are the truth values ofthe statements ∀ x ∀ y ∃ z Q(x, y, z) and ∃ z ∀ x ∀ y Q(x, y, z), wherethe domain of discourse isR3?
What does the statement ∀ x ∀ y ∃ z Q(x, y, z) mean?
‘For all real numbers x and for all real numbers y there is a realnumber z such that it is true that x + y = z’
Is the quantification a true statement? YES
An Example
Example
Let Q(x, y, z) be the statement x + y = z. What are the truth values ofthe statements ∀ x ∀ y ∃ z Q(x, y, z) and ∃ z ∀ x ∀ y Q(x, y, z), wherethe domain of discourse isR3?
What does the statement ∀ x ∀ y ∃ z Q(x, y, z) mean?
‘For all real numbers x and for all real numbers y there is a realnumber z such that it is true that x + y = z’
Is the quantification a true statement? YES
An Example
Example
Let Q(x, y, z) be the statement x + y = z. What are the truth values ofthe statements ∀ x ∀ y ∃ z Q(x, y, z) and ∃ z ∀ x ∀ y Q(x, y, z), wherethe domain of discourse isR3?
What does the statement ∀ x ∀ y ∃ z Q(x, y, z) mean?
‘For all real numbers x and for all real numbers y there is a realnumber z such that it is true that x + y = z’
Is the quantification a true statement? YES
An Example
Example
Let Q(x, y, z) be the statement x + y = z. What are the truth values ofthe statements ∀ x ∀ y ∃ z Q(x, y, z) and ∃ z ∀ x ∀ y Q(x, y, z), wherethe domain of discourse isR3?
What does the statement ∀ x ∀ y ∃ z Q(x, y, z) mean?
‘For all real numbers x and for all real numbers y there is a realnumber z such that it is true that x + y = z’
Is the quantification a true statement?
YES
An Example
Example
Let Q(x, y, z) be the statement x + y = z. What are the truth values ofthe statements ∀ x ∀ y ∃ z Q(x, y, z) and ∃ z ∀ x ∀ y Q(x, y, z), wherethe domain of discourse isR3?
What does the statement ∀ x ∀ y ∃ z Q(x, y, z) mean?
‘For all real numbers x and for all real numbers y there is a realnumber z such that it is true that x + y = z’
Is the quantification a true statement? YES
An Example
What about if we rearrange and look at ∃ z ∀ x ∀ y Q(x, y, z). What isthe meaning here?
‘There is a real number z such that for all real numbers x and for allreal numbers y it is true that x + y = z’.
Is this true or false?
This is false since there is no value for z that satisfies the equationx + y = z for all values of x and y.
An Example
What about if we rearrange and look at ∃ z ∀ x ∀ y Q(x, y, z). What isthe meaning here?
‘There is a real number z such that for all real numbers x and for allreal numbers y it is true that x + y = z’.
Is this true or false?
This is false since there is no value for z that satisfies the equationx + y = z for all values of x and y.
An Example
What about if we rearrange and look at ∃ z ∀ x ∀ y Q(x, y, z). What isthe meaning here?
‘There is a real number z such that for all real numbers x and for allreal numbers y it is true that x + y = z’.
Is this true or false?
This is false since there is no value for z that satisfies the equationx + y = z for all values of x and y.
An Example
What about if we rearrange and look at ∃ z ∀ x ∀ y Q(x, y, z). What isthe meaning here?
‘There is a real number z such that for all real numbers x and for allreal numbers y it is true that x + y = z’.
Is this true or false?
This is false since there is no value for z that satisfies the equationx + y = z for all values of x and y.
Translating Mathematical Statements
ExampleTranslate the statement ‘The sum of two positive integers is alwayspositive’ into a logical expression.
First, what is the domain of discourse?
The domain of discourse is Z × Z .
∀ x ∀ y ((x > 0) ∧ (y > 0)→ (x + y > 0))
Alternately, we can write this as
∀ x ∀ y (x + y > 0)
where the domain of discourse consists of all positive integers.
Translating Mathematical Statements
ExampleTranslate the statement ‘The sum of two positive integers is alwayspositive’ into a logical expression.
First, what is the domain of discourse?
The domain of discourse is Z × Z .
∀ x ∀ y ((x > 0) ∧ (y > 0)→ (x + y > 0))
Alternately, we can write this as
∀ x ∀ y (x + y > 0)
where the domain of discourse consists of all positive integers.
Translating Mathematical Statements
ExampleTranslate the statement ‘The sum of two positive integers is alwayspositive’ into a logical expression.
First, what is the domain of discourse?
The domain of discourse is Z × Z .
∀ x ∀ y ((x > 0) ∧ (y > 0)→ (x + y > 0))
Alternately, we can write this as
∀ x ∀ y (x + y > 0)
where the domain of discourse consists of all positive integers.
Translating Mathematical Statements
ExampleTranslate the statement ‘The sum of two positive integers is alwayspositive’ into a logical expression.
First, what is the domain of discourse?
The domain of discourse is Z × Z .
∀ x ∀ y ((x > 0) ∧ (y > 0)→ (x + y > 0))
Alternately, we can write this as
∀ x ∀ y (x + y > 0)
where the domain of discourse consists of all positive integers.
Translating Mathematical Statements
ExampleTranslate the statement ‘The sum of two positive integers is alwayspositive’ into a logical expression.
First, what is the domain of discourse?
The domain of discourse is Z × Z .
∀ x ∀ y ((x > 0) ∧ (y > 0)→ (x + y > 0))
Alternately, we can write this as
∀ x ∀ y (x + y > 0)
where the domain of discourse consists of all positive integers.
Another Example
ExampleTranslate the statement ‘Every real number except zero has amultiplicative inverse’.
∀ x (x 6= 0)→ ∃ y (xy = 1)
Another Example
ExampleTranslate the statement ‘Every real number except zero has amultiplicative inverse’.
∀ x (x 6= 0)→ ∃ y (xy = 1)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent ‘x is a female‘Let P(x) represent x is a parentLet M(x, y) represent x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent ‘x is a female‘Let P(x) represent x is a parentLet M(x, y) represent x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent
‘x is a female‘Let P(x) represent x is a parentLet M(x, y) represent x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent ‘x is a female‘
Let P(x) represent x is a parentLet M(x, y) represent x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent ‘x is a female‘Let P(x) represent
x is a parentLet M(x, y) represent x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent ‘x is a female‘Let P(x) represent x is a parent
Let M(x, y) represent x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent ‘x is a female‘Let P(x) represent x is a parentLet M(x, y) represent
x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
And Another
ExampleExpress the statement using predicates and quantifiers with a domainof discourse consisting of all people.‘If a person is female and is a parent, then this person is someone’smother’.
First we define the propositional functions. How many do we needhere?
Let F(x) represent ‘x is a female‘Let P(x) represent x is a parentLet M(x, y) represent x is the mother of y
So, our statement would be ...
∀ x ((F(x) ∧ P(x))→ ∃ y M(x, y)
Still Another Example
ExampleExpress the statement ‘Everyone has exactly one best friend’ usingpredicates and quantifiers.
The problem isn’t stating that someone has a best friend, it is statingthat they have exactly one best friend. Thoughts?
We can think of this as ‘For every person x, x has exactly one bestfriend’. To say that x has exactly one best friend implies that there is aperson y who has x as their best friend. Further, for every person zwho is not y then z is not the best friend of y.
Let B(x, y) be the statement ‘x is the best friend of y’.
∀ x ∃ y B(x, y) ∧ ∀ z ((z 6= y)→ ¬B(x, z))
Still Another Example
ExampleExpress the statement ‘Everyone has exactly one best friend’ usingpredicates and quantifiers.
The problem isn’t stating that someone has a best friend, it is statingthat they have exactly one best friend. Thoughts?
We can think of this as ‘For every person x, x has exactly one bestfriend’. To say that x has exactly one best friend implies that there is aperson y who has x as their best friend. Further, for every person zwho is not y then z is not the best friend of y.
Let B(x, y) be the statement ‘x is the best friend of y’.
∀ x ∃ y B(x, y) ∧ ∀ z ((z 6= y)→ ¬B(x, z))
Still Another Example
ExampleExpress the statement ‘Everyone has exactly one best friend’ usingpredicates and quantifiers.
The problem isn’t stating that someone has a best friend, it is statingthat they have exactly one best friend. Thoughts?
We can think of this as ‘For every person x, x has exactly one bestfriend’. To say that x has exactly one best friend implies that there is aperson y who has x as their best friend. Further, for every person zwho is not y then z is not the best friend of y.
Let B(x, y) be the statement ‘x is the best friend of y’.
∀ x ∃ y B(x, y) ∧ ∀ z ((z 6= y)→ ¬B(x, z))
Still Another Example
ExampleExpress the statement ‘Everyone has exactly one best friend’ usingpredicates and quantifiers.
The problem isn’t stating that someone has a best friend, it is statingthat they have exactly one best friend. Thoughts?
We can think of this as ‘For every person x, x has exactly one bestfriend’. To say that x has exactly one best friend implies that there is aperson y who has x as their best friend. Further, for every person zwho is not y then z is not the best friend of y.
Let B(x, y) be the statement ‘x is the best friend of y’.
∀ x ∃ y B(x, y) ∧ ∀ z ((z 6= y)→ ¬B(x, z))
Still Another Example
ExampleExpress the statement ‘Everyone has exactly one best friend’ usingpredicates and quantifiers.
The problem isn’t stating that someone has a best friend, it is statingthat they have exactly one best friend. Thoughts?
We can think of this as ‘For every person x, x has exactly one bestfriend’. To say that x has exactly one best friend implies that there is aperson y who has x as their best friend. Further, for every person zwho is not y then z is not the best friend of y.
Let B(x, y) be the statement ‘x is the best friend of y’.
∀ x ∃ y B(x, y) ∧ ∀ z ((z 6= y)→ ¬B(x, z))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be
‘w has taken f ’Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’
Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’Q(f , a) be
‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Still More Examples
ExampleUse quantifiers to express ‘There is a woman who has taken a flighton every airline in the world’.
Propositional functions?
Let P(w, f ) be ‘w has taken f ’Q(f , a) be ‘f is a flight on a’
What is the domain of discourse?
The domain of discourse would be all flights f , all airlines a and allwomen w.
And the statement.
∃ w ∀ a ∃ f (P(w, f ) ∧ Q(f , a))
Remember Calculus?
ExampleExpress the definition of a limit using quantifiers.
Does anyone remember/know the definition of the statement
limx→a
f (x) = L
For every real number ε > 0, there exists a real number δ > 0 suchthat |f (x)− L| < ε whenever 0 < |x− a| < δ.
So how can we represent this with quantifiers?
∀ ε ∃ δ (0 < |x− a| < δ → |f (x)− L| < ε)
Remember Calculus?
ExampleExpress the definition of a limit using quantifiers.
Does anyone remember/know the definition of the statement
limx→a
f (x) = L
For every real number ε > 0, there exists a real number δ > 0 suchthat |f (x)− L| < ε whenever 0 < |x− a| < δ.
So how can we represent this with quantifiers?
∀ ε ∃ δ (0 < |x− a| < δ → |f (x)− L| < ε)
Remember Calculus?
ExampleExpress the definition of a limit using quantifiers.
Does anyone remember/know the definition of the statement
limx→a
f (x) = L
For every real number ε > 0, there exists a real number δ > 0 suchthat |f (x)− L| < ε whenever 0 < |x− a| < δ.
So how can we represent this with quantifiers?
∀ ε ∃ δ (0 < |x− a| < δ → |f (x)− L| < ε)
Remember Calculus?
ExampleExpress the definition of a limit using quantifiers.
Does anyone remember/know the definition of the statement
limx→a
f (x) = L
For every real number ε > 0, there exists a real number δ > 0 suchthat |f (x)− L| < ε whenever 0 < |x− a| < δ.
So how can we represent this with quantifiers?
∀ ε ∃ δ (0 < |x− a| < δ → |f (x)− L| < ε)
Remember Calculus?
ExampleExpress the definition of a limit using quantifiers.
Does anyone remember/know the definition of the statement
limx→a
f (x) = L
For every real number ε > 0, there exists a real number δ > 0 suchthat |f (x)− L| < ε whenever 0 < |x− a| < δ.
So how can we represent this with quantifiers?
∀ ε ∃ δ (0 < |x− a| < δ → |f (x)− L| < ε)
Translating From Nested Quantifiers into English
ExampleTranslate the statement
∀ x (C(x) ∨ ∃ y (C(y) ∧ F(x, y)))
where C(x) is ‘x has a laptop’, F(x, y) is ‘x and y are friends’ and thedomain of discourse is all students at SSU.
For every student x at SSU, x has a laptop or there is a student y atSSU such that y has a laptop and x and y are friends.
Translating From Nested Quantifiers into English
ExampleTranslate the statement
∀ x (C(x) ∨ ∃ y (C(y) ∧ F(x, y)))
where C(x) is ‘x has a laptop’, F(x, y) is ‘x and y are friends’ and thedomain of discourse is all students at SSU.
For every student x at SSU, x has a laptop or there is a student y atSSU such that y has a laptop and x and y are friends.
Translating From Nested Quantifiers into English
ExampleTranslate the statement
∃ x ∀ y ∀ z ((F(x, y) ∧ F(x, z) ∧ (y 6= z))→ ¬F(y, z))
into English, where F(a, b) means a and b are friends and the domainof discourse for x, y and z consist of all SSU students.
For some SSU student x, for all SSU student y and for all SSU studentz, if x and y are friends, and if x and z are friends and if y and z are notthe same person, then y and z are not friends. In other words, there areno students whose friends are also friends.
Translating From Nested Quantifiers into English
ExampleTranslate the statement
∃ x ∀ y ∀ z ((F(x, y) ∧ F(x, z) ∧ (y 6= z))→ ¬F(y, z))
into English, where F(a, b) means a and b are friends and the domainof discourse for x, y and z consist of all SSU students.
For some SSU student x, for all SSU student y and for all SSU studentz, if x and y are friends, and if x and z are friends and if y and z are notthe same person, then y and z are not friends. In other words, there areno students whose friends are also friends.
Negating Quantifiers
ExampleNegate the statement
∀ x ∃ y (xy = 1)
There are three different ways we can express this:
1 ∃ x ¬∃ y (xy = 1)2 ∃ x ∀ y ¬(xy = 1)3 ∃ x ∀ y (xy 6= 1)
Negating Quantifiers
ExampleNegate the statement
∀ x ∃ y (xy = 1)
There are three different ways we can express this:
1 ∃ x ¬∃ y (xy = 1)2 ∃ x ∀ y ¬(xy = 1)3 ∃ x ∀ y (xy 6= 1)
Negating Quantifiers
ExampleNegate the statement
∀ x ∃ y (xy = 1)
There are three different ways we can express this:
1 ∃ x ¬∃ y (xy = 1)
2 ∃ x ∀ y ¬(xy = 1)3 ∃ x ∀ y (xy 6= 1)
Negating Quantifiers
ExampleNegate the statement
∀ x ∃ y (xy = 1)
There are three different ways we can express this:
1 ∃ x ¬∃ y (xy = 1)2 ∃ x ∀ y ¬(xy = 1)
3 ∃ x ∀ y (xy 6= 1)
Negating Quantifiers
ExampleNegate the statement
∀ x ∃ y (xy = 1)
There are three different ways we can express this:
1 ∃ x ¬∃ y (xy = 1)2 ∃ x ∀ y ¬(xy = 1)3 ∃ x ∀ y (xy 6= 1)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack
∀ x F(x, Jack)2 Owen can fool everybody∀ x F(Owen, x)
3 Everybody can fool somebody∀ x ∃ y F(x, y)
4 There is no one who can fool everybody∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack∀ x F(x, Jack)
2 Owen can fool everybody∀ x F(Owen, x)
3 Everybody can fool somebody∀ x ∃ y F(x, y)
4 There is no one who can fool everybody∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack∀ x F(x, Jack)
2 Owen can fool everybody
∀ x F(Owen, x)3 Everybody can fool somebody∀ x ∃ y F(x, y)
4 There is no one who can fool everybody∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack∀ x F(x, Jack)
2 Owen can fool everybody∀ x F(Owen, x)
3 Everybody can fool somebody∀ x ∃ y F(x, y)
4 There is no one who can fool everybody∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack∀ x F(x, Jack)
2 Owen can fool everybody∀ x F(Owen, x)
3 Everybody can fool somebody
∀ x ∃ y F(x, y)4 There is no one who can fool everybody∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack∀ x F(x, Jack)
2 Owen can fool everybody∀ x F(Owen, x)
3 Everybody can fool somebody∀ x ∃ y F(x, y)
4 There is no one who can fool everybody∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack∀ x F(x, Jack)
2 Owen can fool everybody∀ x F(Owen, x)
3 Everybody can fool somebody∀ x ∃ y F(x, y)
4 There is no one who can fool everybody
∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can fool Jack∀ x F(x, Jack)
2 Owen can fool everybody∀ x F(Owen, x)
3 Everybody can fool somebody∀ x ∃ y F(x, y)
4 There is no one who can fool everybody∀ x ∃ y ¬F(x, y)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody
∀ x ∃ y F(y, x)2 No one can fool both Jack and Owen∀ x (F(x,Owen)⊕ F(x, Jack))
3 Jack can fool exactly two people∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself∀ x ¬F(x, x)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody∀ x ∃ y F(y, x)
2 No one can fool both Jack and Owen∀ x (F(x,Owen)⊕ F(x, Jack))
3 Jack can fool exactly two people∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself∀ x ¬F(x, x)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody∀ x ∃ y F(y, x)
2 No one can fool both Jack and Owen
∀ x (F(x,Owen)⊕ F(x, Jack))3 Jack can fool exactly two people∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself∀ x ¬F(x, x)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody∀ x ∃ y F(y, x)
2 No one can fool both Jack and Owen∀ x (F(x,Owen)⊕ F(x, Jack))
3 Jack can fool exactly two people∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself∀ x ¬F(x, x)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody∀ x ∃ y F(y, x)
2 No one can fool both Jack and Owen∀ x (F(x,Owen)⊕ F(x, Jack))
3 Jack can fool exactly two people
∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself∀ x ¬F(x, x)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody∀ x ∃ y F(y, x)
2 No one can fool both Jack and Owen∀ x (F(x,Owen)⊕ F(x, Jack))
3 Jack can fool exactly two people∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself∀ x ¬F(x, x)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody∀ x ∃ y F(y, x)
2 No one can fool both Jack and Owen∀ x (F(x,Owen)⊕ F(x, Jack))
3 Jack can fool exactly two people∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself
∀ x ¬F(x, x)
One Final (Long) Example
Example
Let F(x, y) be the statement ‘x can fool y’, where the domain ofdiscourse consists of all people in the world. Use quantifiers toexpress each of the following statements:
1 Everybody can be fooled by somebody∀ x ∃ y F(y, x)
2 No one can fool both Jack and Owen∀ x (F(x,Owen)⊕ F(x, Jack))
3 Jack can fool exactly two people∃ x ∃ y (x 6= y ∧ F(Jack, x) ∧ F(Jack, y) ∧ ∀ z (F(Jack, z)→(z = x ∨ z = y)))
4 No one can fool themself∀ x ¬F(x, x)