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16+Introduction+to+Relations

# 16+Introduction+to+Relations - Handout#16 CS103 Robert...

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Handout #16 CS103 April 13, 2011 Robert Plummer Introduction to Relations Relations are a fundamental concept in discrete mathematics, used to define how sets of objects relate to other sets of objects. Not only do they provide a formal way of being able to talk about such relationships, they also provide the most widespread model used in modern commercial database systems. Understanding relations from a mathematical perspective not only gives you an important modeling tool, but also gives you the foundational theory used in a number of applications including relational database management systems, task scheduling systems, and methods to solve various optimization problems. To explore what relations are, let’s begin by considering the following set G Greek deities: G = {Zeus, Apollo, Cronus, Poseidon} As you may know, Zeus is the father of Apollo, Cronus is the father of Poseidon, and Cronus is also the father of Zeus. So some of the elements of G that satisfy the "is the father of" relation with respect to others Notice that in this case, the elements that are related are both from the same set, G, and that the relationship is "one way": if X is the father of Y, then Y is not the father of X. If we had another set, H, of female deities, then some members of G might bear the "is married to" relation to members of H, and that relationship would also be true in the other direction. We will formalize all of these ideas, starting with some definitions. Definitions A sequence of objects is a list of these objects in some order. Sequences may be finite or infinite. A finite sequence is called a tuple . A sequence with k objects is called a k-tuple . An ordered pair is a 2-tuple; that is, an ordered sequence of two elements. We write ordered pairs in parentheses, for example (a, b) , and we call a the first element and b the second element of the pair. The Cartesian product or cross product of two sets A and B, written A B , is the set of all ordered pairs wherein the first element is a member of A and the second element is a member of B. A binary relation R between two sets A and B (which may be the same) is a subset of the Cartesian product A B . If element a A is related by R to element b B, we denote this fact by writing (a, b) R , or alternately, by a R b . We say that R is a relation on A and B . A relation on a set A is a subset of A A.

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– 2 – A good way to think of a binary relation is that it is a way to designate that of all the ordered pairs in the cross product of two sets, some are "interesting" because there is a certain relationship between them. We often name relations with capital letters, but some relations, such as "less- than" have their own symbols, like "<". What we defined above is a binary relation because it operates on ordered pairs. We can also define unary relations, which operate on single elements, or ternary relations , which operate on ordered triples. In general an n -ary relation will operate on n -tuples. Formally, we can express this as: Definition An n -ary relation on the sets A 1 , A 2 , …, A n
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