2.2N_Multiple_Quantifiers

2.2N_Multiple_Quantifiers - Math 280 Strategies of Proof...

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Unformatted text preview: Math 280 Strategies of Proof Fall 2007 Class Notes 2.2 Multiply Quantified Statements Propositional functions may (and often do) involve more than one variable. For a propositional function with multiple variables, we may apply a quantifier to each variable. For instance, suppose p ( x, y ) is a propositional function involving two variables x and y . We may apply quantifiers to both variables, as in ∀ x ∀ y p ( x, y ) or ∀ x ∃ y p ( x, y ) . The result in either case will be a statement. If we have the same quantifier on both variables, then the order of the quantifiers does not matter. The statement For all x , for all y , p ( x, y ). is logically equivalent to the statement For all y , for all x , p ( x, y ). In symbolic form, this means ∀ x ∀ y p ( x, y ) ≡ ∀ y ∀ x p ( x, y ) . We usually read either of these statements more concisely as For all x and y , p ( x, y ). A similar result holds for two existential quantifiers: ∃ x ∃ y p ( x, y ) ≡ ∃ y ∃ x p ( x, y ) . These statements are usually read There exist x and y such that p ( x, y ). Example. Rewrite the following statements in symbolic form. (a) For all real numbers x and y , x + y = y + x . (This is a statement of the commutative property of addition for real numbers.) Solution. Define the propositional function p ( x, y ) : x + y = y + x. The statement then has symbolic form ∀ x ∀ y p ( x, y ) . Alternatively, we can just include the propositional function in the quantified statement with- out using a separate notation. The symbolic form is then ∀ x ∀ y [ x + y = y + x ] . The domain for both x and y is the set R of real numbers. (b) For all real numbers x , y , and z , ( x + y ) + z = x + ( y + z ). (This is a statement of the associative property of addition for real numbers.) Solution. Define the propositional function q ( x, y, z ) : ( x + y ) + z = x + ( y + z ) . The statement then has symbolic form ∀ x ∀ y ∀ z q ( x, y, z ) or alternatively, ∀ x ∀ y ∀ z [( x + y ) + z = x + ( y + z )] . The domain for each of x , y , and z is the set R of real numbers. 45 46 Chapter 2 Predicate Logic Example. (continued) (c) There exist odd prime numbers m and n such that m + n = 100. (This is a particular case of what is called Goldbach’s Conjecture.) Solution. Define the propositional function r ( m, n ) : m + n = 100 . The statement then has symbolic form ∃ m ∃ n r ( m, n ) or alternatively, ∃ m ∃ n [ m + n = 100] . The domain for both m and n is the set of odd prime numbers. Mixed Quantifers When a quantified statement involves both existential and universal quantifiers, or mixed quanti- fiers , then the order of the quantifiers does matter for both the meaning of the statement and in its truth value. For instance, suppose p ( x, y ) is a propositional function involving the variables x and y . Then the statement ∀ x ∃ y p ( x, y ) is read For all x , there exists y such that p ( x, y )....
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2.2N_Multiple_Quantifiers - Math 280 Strategies of Proof...

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