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Set Theory and Predicate Logic
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Ordered Pairs, Tuples, and Cartesian Products
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An important concept in mathematics and one that is central to database programming is that of an ordered pair (a,b). To include ordered pairs in a rigorous treatment of mathematics, there must be a universal set of ordered pairs. This is the Cartesian product.
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Ordered Pairs and k-Tuples
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We will consider ordered pair to be a new unde ned term, like set. Recall that a particular set is de ned by its members and nonmembers; a particular ordered pair is de ned by its rst part and its second part. We also accept without de nition the term tuple, or k-tuple, for an object that, like an ordered pair, has parts but where there are k parts. An ordered pair is a tuple in particular, a 2-tuple; (x,y,z,w) is also a tuple and, in particular, a 4-tuple.
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If s and t are elements of some domains, (s,t) is called the ordered pair with rst part (or coordinate) s and second part (or coordinate) t. Two ordered pairs (s,t) and (x,y) with matching domains are equal if their corresponding parts are equal: s=x and t=y. If s, t, . . ., r are (k-many) elements of some domains, (s,t,. . .,r) is called an ordered k-tuple. Reference to the parts of (s,t,. . .,r) and equality for k-tuples follow the analogues for ordered pairs. Subscript notation is used for the parts of ordered pairs and tuples, when the tuple itself is represented by a single symbol. It s especially convenient when all the parts have a common domain. If r is a k-tuple of real numbers and j is an integer between 1 and k, rj is a real number and denotes the jth part of r. The most familiar example of ordered pairs in mathematics, and perhaps the original one, is the usual notation for points in the coordinate plane: (x,y), where x and y are real numbers. The seventeenth-century mathematician Ren Descartes used this notation, which is now called the Cartesian coordinate system in his honor. Naming the points in the plane (x,y) works. In the sense we described earlier, this notation faithfully represents the essence of points. Thus, nothing is lost by saying the point (x,y) instead of the point represented by (x,y). The set of all points in the plane is P = {(x,y) : x and y }. A more compact way to write the set P is , which mathematicians understand to mean the same thing and which is called the Cartesian product of and .
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A Cartesian product is the domain of discourse for ordered pairs or tuples. Here s the general de nition.
De nition of Cartesian Product
Let S and T be sets. The Cartesian product of S and T, denoted S T, is the set {(s,t) : s S and t T}. If no confusion arises, the terms S-coordinate and T-coordinate can be used in place of rst coordinate and second coordinate, respectively, for the parts of elements of S T. The sets S and T are called factors (and if needed, the rst and second factors, respectively) of S T. Cartesian products with more than two factors are de ned analogously as sets of tuples, with no distinction made between, for example, (A B) C, which contains elements of the form ((a,b),c), and A B C, which contains elements of the form (a,b,c).
Note In the de nitions for ordered pairs, equality of ordered pairs was de ned as coordinate-wise
equality on the coordinate parts. Any operation de ned on a Cartesian product s factors can similarly be lifted, or imparted to the elements of S T. When this is done, the operation is said to be a coordinate-wise operation. In the Cartesian plane, for example, a coordinate-wise + operation combines the points (x,y) and (s,t) to obtain (x+s,y+t). With the exception of the = operator, don t assume a familiar symbol represents a coordinate-wise operation on ordered pairs (or tuples). For example, although |s| means the absolute value of the number s, |(s,t)| does not represent (|s|,|t|). The Cartesian product is not commutative: A B and B A are not the same when A and B are different.
The Empty Set(s)
The empty set contains no elements, but what is its universe If imagining a set of all sets gets us into trouble, a set of all possible elements can only be worse because sets can be elements of sets. As we ve seen before, using the word the doesn t make something unique. The empty set of integers is the set whose elements are (there are none) and whose nonelements comprise all integers. On the other hand, the empty set of English words is the set whose elements are (there are none) and whose nonelements comprise all English words. How many empty sets are there Perhaps my insistence that sets have well-de ned domains has back red and buried us in empty sets! Fortunately, we can declare the question invalid. Our framework only de nes equality of things and questions such as how many within some universal set U, and no universal set contains all the empty sets. We do want to know how to interpret any sentence containing the phrase the empty set, and that we can know.
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