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order, ending up with handled(e,o) for employee e handled order o. That s not wrong, and in fact it might be the best way to begin if we were designing a database to support queries to return S. To de ne S mathematically, however, the three-detail notion is closer to what de nes S as a set of customers: whether a particular customer c is in the set S. It s harder to express S mathematically with the two-detail interpretation.
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The last element in the description we need notation for is from the USA. Being from the USA or not is a property of employees, and we ll write fromUSA(e) to represent the truth value of employee e is from the USA. To make things a bit simpler to write down at rst, let USAEmployees be the set of employees from the USA or, mathematically, let USAEmployees = {e Employees : fromUSA(e)}. Now that we ve named everything we might need, we turn to the question of describing membership in S in terms of the objects we ve de ned. Question In terms of the sets Customers, USAEmployees, and Orders and the function handled(e,o,c), when is a particular customer c in S Answer The customer c is in S if and only if for every (employee) e in the set USAEmployees, there is at least one (order) o in the set Orders for which handled(e,o,c).
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2
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Set Theory and Predicate Logic
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Using mathematical notation only, here s what we get:
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De nition of the Set S (in Mathematics)
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Let USAEmployees = {e S = {c Customers : Employees : fromUSA(e)}. Then de ne the set e USAEmployees ( o Orders : (handled(e,o,c)))}
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At the end of this chapter, we ll revisit this set.
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Well-De nedness
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In nonmathematical language, we describe something as well-de ned if it has a distinct boundary or outline. In mathematics, well-de ned has a different meaning. Mathematicians call something well-de ned if it s de ned unambiguously. Read the following terms and descriptions and decide which terms are de ned unambiguously. Provinces The set of Canadian provinces Numerator Low Temp Big Number The numerator of the number 0.2 written as a fraction The lowest temperature ever recorded in Russia The largest number that can be described with fewer than 20 words
Contact List The name and a phone number for each of this book s authors, alphabetized by author s last name Shortest Book Square x2 Letter The letter B The book in the Library of Congress that has the fewest pages
Let s see if we agree on which of these are well-de ned. Provinces This is a well-de ned set: One way of denoting this set is {Alberta, British Columbia, Manitoba, New Brunswick, Newfoundland and Labrador, Nova Scotia, Ontario, Prince Edward Island, Quebec, Saskatchewan}. Numerator This number isn t well-de ned because we have many ways to write 0.2 as a fraction, and they don t all have the same numerator. Low Temp This is well-de ned, even though we might not know the value.
Big Number Although this may appear to be a valid de nition, it s not. Consider the number N plus one, where N is the largest number that can be described with fewer than 20 words. This is a variation on the Berry Paradox.
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Contact List This isn t well-de ned if any of the authors has more than one phone number because it doesn t specify how we choose phone numbers for the list. Shortest Book Although the minimum number of pages is well-de ned (assuming a standard procedure for counting pages), more than one book might have the minimum number of pages. As a result, we can t be sure there is a single shortest book. Square We don t know the value of x, so x2 isn t a well-de ned number. On the other hand, it is a well-de ned algebraic expression. Letter This de nes a particular letter of the English alphabet but not a speci c example of that letter in, say, a copy of this book. These simple examples offer a number of lessons, but I ll mention just one in particular: English can easily mislead. For example, two words that indicate uniqueness the de nite article the and the superlative shortest were used to describe something that wasn t in fact unique. Later in this chapter, I ll be more speci c about the notion of well-de nedness as it applies to sets.
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