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Unformatted text preview: 1 Universal Quantifiers: Specialization 1.1 Welcome Welcome to the sixth module in our course on proofs, “Universal Quantifiers: Specialization.” Before going on, please do the assigned readings. 1.2 Introduction The previous module introduced the choose method to work backwards from a statement containing a universal quantifier. This module introduces the specialization method to work forwards from a statement containing a universal quantifier. 1.3 When To Use Specialization We should consider using specialization whenever two conditions are met. 1. A universal quantifier appears in the hypothesis or forward process. 2. A particular object is identified in the conclusion or the backward process. Consider the following example. We begin with a definition. Let S be a set of real numbers. An upper bound for S is a real number u such that, for all x ∈ S , x ≤ u . Proposition 1. Let u and v be real numbers. If u is an upper bound for a set S and u ≤ v , then v is an upper bound for the set S . Since a universal quantifier is used implicitly in the statement of the proposition, as part of the definition of upper bound, and because a particular object v is identified in the conclusion, this is a natural setting for specialization....
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- Spring '08
- Complex number, upper bound, Use specialization