1. 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

2. 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

3. 3.3. Statements with Multiple Quantifiers
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.
P(Tokyo, Japan)
P(Athens, Egypt)
P(Paris, France)
P(Miami, Brazil)
3.3. Statements with Multiple Quantifiers

4. 3.3. Statements with Multiple Quantifiers
Example – pg. 129 # 2
Let G(x, y) be “x2 > y.” Indicate which of the following statements are true and which are false.
G(2, 3)
G(1, 1)
G(1/2, 1/2)
G(-2, 2)
3.3. Statements with Multiple Quantifiers

5. Negations of Multiply-Quantified Statements
3.3. Statements with Multiple Quantifiers

6. 3.3. Statements with Multiple Quantifiers
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.
19.
3.3. Statements with Multiple Quantifiers

7. 3.3. Statements with Multiple Quantifiers
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.
3.3. Statements with Multiple Quantifiers