Discrete Structures
Chapter 3: The Logic of Quantified Statements
3.3. Statements with Multiple Quantifiers
It is not enough to have a good mind. The main thing is to use it
well.
– René Descartes, 1596 – 1650
3.3. Statements with Multiple Quantifiers
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Multiple Quanitifiers
• We begin by considering sentences in which
there is more than one quantifier of the same
“quantity”—i.e., sentences with two or more
existential quantifiers, and sentences with two
or more universal quantifiers.
3.3. Statements with Multiple Quantifiers
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Example – pg. 129 # 1
• Let C be the set of cities in the world, let N be
the set of nations in the world, and let P(c, n)
be “c is the capital city of n.” Determine the
truth values of the following statements.
a.
b.
c.
d.
P(Tokyo, Japan)
P(Athens, Egypt)
P(Paris, France)
P(Miami, Brazil)
3.3. Statements with Multiple Quantifiers
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Example – pg. 129 # 2
• Let G(x, y) be “x2 > y.” Indicate which of the
following statements are true and which are
false.
a.
b.
c.
d.
G(2, 3)
G(1, 1)
G(1/2, 1/2)
G(-2, 2)
3.3. Statements with Multiple Quantifiers
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Negations of Multiply-Quantified
Statements
•
~  x in D, y in E s.t. P  x, y    x in D s.t. y in E , ~ P  x, y  .
• ~  x in D s.t. y in E , P  x, y    x in D, y in E s.t. ~P  x, y  .
3.3. Statements with Multiple Quantifiers
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Example – pg. 129 # 13 & 19
• In each case, (a) rewrite the statement in
English without using the symbol  or  or
variables and expressing your answer as
simply as possible, and (b) write a negation for
the statement.
– 13.  colors C,  an animal A s.t. A is colored C.
– 19. x 
s.t. for all real numbers y, x  y  0.
3.3. Statements with Multiple Quantifiers
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Example – pg. 131 # 56
• Let P(x) and Q(x) be predicates and suppose D is the domain
of x. For the statement forms in each pair, determine whether
(a) they have the same truth value for every choice of P(x),
Q(x), and D, or (b) there is a choice of P(x), Q(x), and D for
which they have opposite truth values.
x  D,  P  x   Q  x  
and
 x  D, P  x    x  D, Q  x 
3.3. Statements with Multiple Quantifiers
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