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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 • • • • • 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.