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Unformatted text preview: 1 Universal Quantifiers: The Choose Method 1.1 Welcome Welcome to the fifth module in our course on proofs, “Universal Quantifiers: The Choose Method”. Before going on, please do the assigned readings. 1.2 Introduction The previous module dealt with the existential quantifier, there exists . This module deals with the universal quantifier, for all . 1.3 Basic Structure and Examples All statements which use universal quantifiers share a basic structure, though some elements of the structure may be absent, implicit or appear in a different order. For every “object” in the “universe of discourse” with a “certain property”, “something happens”. It is very important to be able to identify the object, the universe of discourse, the property and the something that happens. As with existential quantifiers, the universe of discourse is the set from which “objects” are taken. It is the domain of the objects. Here are some examples. 1. For every integer n > 5, 2 n > n 2 . Object: n Universe of discourse: Z Certain property: n > 5 Something happens: 2 n > n 2 The statement might appear as “2 n > n 2 for all integers n > 5”. The order is different but the meaning is the same. 2. For every angle θ , sin 2 ( θ ) + cos 2 ( θ ) = 1. Object: θ Universe of discourse: R , inferred from context Certain property: None specified. Something happens: sin 2 ( θ ) + cos 2 ( θ ) = 1 3. For all continuous, real-valued functions f on [ a,b ] with f ( a ) > 0 and f ( b ) < 0, there exists a real number c ∈ ( a,b ) such that f ( c ) = 0. This statement is more complex. Not only does the statement begin with a universal quantifier, but the “something that happens” contains an existential quantifier. 1 Object: f Universe of discourse: continuous, real-valued functions on [ a,b ] Certain property: f ( a ) > 0, f ( b ) < Something happens: ∃ c ∈ ( a,b ) 3 f ( c ) = 0 It takes practice to become fluent in reading and writing statements that use quantifiers. 1.4 Notation The mathematical symbol for for all (and its equivalent forms) is ∀ . Here is a table of statements using universal quantifiers. The left hand column is wholly symbolic. The right hand column is more verbose and is more commonly used. Notice that in the first two cases, simply reading the mathematical expression from left to right produces the more natural sounding English sentence. In the last case, for all is implied in the English language sentence. Symbolic Verbose ∀ n ∈ Z ,n > 5, 2 n > n 2 For every integer n > 5, 2 n > n 2 . ∀ ∠ θ , sin 2 ( θ ) + cos 2 ( θ ) = 1 For every angle θ , sin 2 ( θ ) + cos 2 ( θ ) = 1 ∀ n ∈ P , ∑ n i =1 i = n ( n + 1) / 2 The sum of the first n positive integers is n ( n + 1) / 2....
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- Spring '08
- Empty set, Diophantine equation, certain property