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2.2.1
Section 2 - More Quantified
Statements
• Statements with multiple quantifiers;
• Negations of multiply quantified statements;
• Equivalent forms of universal conditionals.
2.2.2
Multiply Quantified Statements
• Consider the following statement:
Given any real number, there is a
smaller real number.
• This is equivalent to the formal statement:
∀ x∈R, ∃ y∈R ∋ y < x.
• This is an example of a multiply quantified
statement.
2.2.3
Examples
• The formal statement:
∃ x∈R+ ∋ ∀ y∈R+, y < x
can be interpreted informally as:
• There is a non-negative real number with the
property that all other non-negative real
numbers are smaller than this number;
• There is a non-negative real number that is
larger than all other non-negative real numbers.
2.2.4
Another Example
• INFORMAL:
Everybody loves somebody.
• FORMAL:
∀ people x, ∃ a person y ∋ x loves y.
• INFORMAL:
Somebody loves everybody.
• FORMAL:
∃ a person x ∋ ∀ people y, x loves y.
2.2.5
Negation of Universal Existentials
• What is the negation of the statement:
∀ people x, ∃ a person y such that x loves y?
• Recall this is “Everybody loves somebody,” so
its negation would be the case of “Somebody
who does not love anybody.”
• In formal terms:
∃ a person x ∋ ∀ people y, x does not love y.
• Thus:
~ [∀ x, ∃ y ∋ P(x,y) ] ≡ ∃ x ∋ ∀ y, ~P(x,y)
2.2.6
Negation of Existential Universals
• What is the negation of the statement:
∃ a person x such that ∀ people y, x loves y?
• Recall this is “Somebody loves everybody,” so
its negation would be the case of “Everybody
has at least one person they do not love.”
• In formal terms:
∀ people x, ∃ person y ∋ x does not love y.
• Thus:
~ [∃ x ∋ ∀ y, P(x,y) ] ≡ ∀ x, ∃ y ∋ ~P(x,y)
Equivalent Forms of
Universal Conditionals
2.2.7
• Given the statement:
∀ x∈D, if P(x), then Q(x)
analogous to our definitions from propositional
calculus, we can construct the following.
• Contrapositive: ∀ x∈D, if ~Q(x), then ~P(x).
• Converse: ∀ x∈D, if Q(x), then P(x).
• Inverse: ∀ x∈D, if ~P(x), then ~Q(x).
• Negation: ∃ x∈D ∋ P(x), and ~Q(x).
2.2.8
Example
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Statement:
∀ x∈R, if x > 2, then x2 > 4.
Converse:
∀ x∈R, if x2 > 4, then x > 2.
Inverse:
∀ x∈R, if x ≤ 2, then x2 ≤ 4.
Contrapositive: ∀ x∈R, if x2 ≤ 4, then x ≤ 2.
Negation:
∃ x∈R ∋ x > 2 and/but x2 ≤ 4.