Section 1.7
More with Quantifiers
DeMorgan’s Law for Quantified Statements
Announcements
• Don’t forget to pay attention to the actual
reading schedule.
• Phones should be away unless we are using
them in class.
• You are supposed to be participating with
your groups during the group activities.
Activity #1
• Assume:
–
–
–
–
the domain consists of integers
O(x) is “x is odd”
L(x) is “x < 10”
G(x) is “x>9”
Activity #1
• Given the prior assumptions, what is the
truth value of the following statements.
1.
2.
3.
4.
5.
 x [ O(x) ]
 x [L(x)  O(x) ]
 x [L(x)  ¬ G(x) ]
 x [L(x)  G(x)]
 x [L(x)  G(x)
Activity #1
• Assume:
– the domain consists of integers
– A(x) is “x<5”
– B(x) is “x<7”
Activity #1
• Given the prior assumptions, what is the
truth value of the following statements.
1.
2.
3.
4.
5.
 x [ A(x) ]
 x [ A(x)  B(x)]
 x [ A(x)  B(x)]
 x [ A(x)  B(x) ]
 x [ B(x)  A(x) ]
Activity #2
• How would you write the negation of:
– All Americans eat cheeseburgers
– There is a smart student at the University of
Iowa.
Negation of a Quantified Statement
• You had to pay careful attention to negation
with propositional logic (DeMorgan’s
laws).
– Predicate Logic has the same need for detail
Negation of a Quantified Statement
• Consider the following statement:
– Every student in this course has taken a class in
statistics.
• This is a standard universal quantification:
 x  D, Q(x)
• What does this look like if I negate the
statement:
¬ (  x  D, Q(x) )
Negation of a Quantified Statement
• Actually, they are called DeMorgan’s Laws
for Quantifiers, and they LOOK like
DeMorgan’s Laws for Propositional Logic
¬ (  x  D, Q(x) )   x  D, ¬ Q(x)
¬ (  x  D, Q(x) )   x  D, ¬ Q(x)
Negation of a Universally Quantified
Statement
• Thus, the negation of a universal statement
(“all are”) is logically equivalent to an
existential statement with negation
¬ (  x  D, Q(x) )   x  D, ¬ Q(x)
(“some are not” or “there is at least one that is not”)
Negation of a Universally
Quantified Statement
• Think about this. If we say
 x  D, Q(x) is false
• What we are saying is:
¬ [  x  D, Q(x) ]
Negation of a Universally
Quantified Statement
¬ [  x  D, Q(x) ]
¬ [ Q(x1)  Q(x2)  Q(x3)  …  Q(xn) ]
Equivalent expression/definition of the Universal Statement
¬ Q(x1)  ¬ Q(x2)  ¬ Q(x3)  …  ¬ Q(xn)
DeMorgan’s Law of negating conjunction
 x  D, ¬ Q(x)
Equivalent expression of the Existential Statement
Negation of an Existentially
Quantified Statement
• Thus, the negation of an existential
statement (“some are”) is logically
equivalent to a universal statement with
negation
¬ (  x  D, Q(x) )   x  D, ¬ Q(x)
(“none are” or “all are not”)
Similarly
 x  D, Q(x) is false
¬ [ x  D, Q(x) ]
¬ [ Q(x1)  Q(x2)  Q(x3)  …  Q(xn) ]
Equivalent expression/definition of the Existential Statement
¬ Q(x1)  ¬ Q(x2)  ¬ Q(x3)  …  ¬ Q(xn)
DeMorgan’s Law of negating disjunction
 x  D, ¬ Q(x)
Equivalent expression of the Universal Statement
Negation of a Quantified Statement
• Thus, the negation of
Every student in this course has taken a class in statistics.
• Is not:
No student in this class has taken a class in statistics
• But
There exists at least one student in this class who has not
taken statistics
Activity #2
• How would you write the negation of:
– All Americans eat cheeseburgers
– There is a smart student at the University of
Iowa.
Negations of Universal
Conditional Statements
• The negation of a universal conditional
statement is:
¬ (  x, if P(x) then Q(x) ) 
 x such that P(x) and ¬ Q(x)
Activity #3
•  real numbers x, if x2 >= 1 then x> 0
• For every student at UNI if they have been
at UNI for at least two years then they are
classified as a Junior.
Vacuous Truth of Universal Statement
• Every senior citizen in this room fought
during world war II.
• Is this a true statement or a false statement?
Vacuous Truth of Universal Statement
• To be FALSE then it’s negation must be TRUE
– That’s the definition of negation.
• But the negation of that statement is:
– There exists at least one senior citizen in this room that
did not fight in world war II.
– This statement is clearly false (since there are no senior
citizens in this room for which this is true)
– This means that the original statement MUST be true