s7_1 - CHAPTER 7 Introduction to Relations 1 Relations and...

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CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition : A binary relation from a set A to a set B is a subset R A × B. If ( a, b ) R we say a is related to b by R . A is the domain of R , and B is the codomain of R . If A = B , R is called a binary relation on the set A . Notation : If ( a, b ) R , then we write aRb . If ( a, b ) 6∈ R , then we write a 6 R b . Discussion Notice that a relation is simply a subset of A × B . If ( a, b ) R , where R is some relation from A to B , we think of a as being assigned to b . In these senses students often associate relations with functions. In fact, a function is a special case of a relation as you will see in Example 1.2.4. Be warned, however, that a relation may differ from a function in two possible ways. If R is an arbitrary relation from A to B , then it is possible to have both ( a, b ) R and ( a, b 0 ) R , where b 0 6 = b ; that is, an element in A could be related to any number of elements of B ; or it is possible to have some element a in A that is not related to any element in B at all. 204
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1. RELATIONS AND THEIR PROPERTIES 205 Often the relations in our examples do have special properties, but be careful not to assume that a given relation must have any of these properties. 1.2. Examples. Example 1.2.1 . Let A = { a, b, c } and B = { 1 , 2 , 3 , 4 } , and let R 1 = { ( a, 1) , ( a, 2) , ( c, 4) } . Example 1.2.2 . Let R 2 N × N be defined by ( m, n ) R 2 if and only if m | n . Example 1.2.3 . Let A be the set of all FSU students, and B the set of all courses offered at FSU. Define R 3 as a relation from A to B by ( s, c ) R 3 if and only if s is enrolled in c this term. Discussion There are many different types of examples of relations. The previous examples give three very different types of examples. Let’s look a little more closely at these examples. Example 1.2.1. This is a completely abstract relation. There is no obvious reason for a to be related to 1 and 2. It just is. This kind of relation, while not having any obvious application, is often useful to demonstrate properties of relations. Example 1.2.2. This relation is one you will see more frequently. The set R 2 is an infinite set, so it is impossible to list all the elements of R 2 , but here are some elements of R 2 : (2 , 6) , (4 , 8) , (5 , 5) , (5 , 0) , (6 , 0) , (6 , 18) , (2 , 18) . Equivalently, we could also write 2 R 2 6 , 4 R 2 8 , 5 R 2 5 , 5 R 2 0 , 6 R 2 0 , 6 R 2 18 , 2 R 2 18 . Here are some elements of N × N that are not elements of R 2 : (6 , 2) , (8 , 4) , (2 , 5) , (0 , 5) , (0 , 6) , (18 , 6) , (6 , 8) , (8 , 6) . Example 1.2.3. Here is an element of R 3 : (you, MAD2104). Example 1.2.4 . Let A and B be sets and let f : A B be a function. The graph of f , defined by graph ( f ) = { ( x, f ( x )) | x A } , is a relation from A to B . Notice the previous example illustrates that any function has a relation that is associated with it. However, not all relations have functions associated with them.
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  • Fall '08
  • PenelopeKirby
  • Binary relation, Transitive relation, Symmetric relation, Composition of relations, Inverse relation, Transitive Relations

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