D-Math-W05-Ch3 - SHEN'S CLASS NOTES-191 Chapter Three...

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SHEN’S CLASS NOTES-191 1 Chapter Three Relations 3.1 Relations Relations generalize the notion of functions. Definition 3.1 A (binary) relation R from a set X to a set Y is a subset of the Cartesian product X × Y . If ( x , y ) R , we say that x is related to y and write x R y . If X = Y , we call R a (binary) relation on X . Example 1 (3.1.3) Let X = {2, 3, 4} and Y = {3, 4, 5, 6, 7}. We define a relation R as follows. R = {( x , y ) | x X , y Y , x divides y }. Then, R = {(2, 4), (2, 6), (3, 3), (3, 6), (4, 4)}. A binary relation can also be represented by a table or an arrow diagram as shown in Fig. 1. From the figure, we can see that a relation does not require that ( x , y ) must be unique for each x . A function is a special case of a relation.
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SHEN’S CLASS NOTES-191 2 X Y 2 4 2 6 3 3 3 6 4 4 (a) Table representation 2 3 4 3 4 5 6 7 (b) Arrow diagram Fig. 1 Table and arrow diagram representation of a relation. Example 2 (3.1.4) Let X = {1, 2, 3, 4}. We define a relation on X by ( x , y ) R if x y . Then R ={(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)}. A relation on set X can be represented by a digraph (a directed graph) G( X , E ), where ( x , y ) E if and only if ( x , y ) R . For the above relation, its digraph is shown in Fig. 2. 3 4 1 2 Fig. 2 Digraph representation of a relation.
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SHEN’S CLASS NOTES-191 3 Definition 3.2 A relation on a set X is called reflexive if ( x , x ) R for every x X . The relation in Example 2 (Fig. 2) is reflexive. Example 3 (3.1.7) Let X = { a , b , c , d }. The following relation R is not reflexive. R = {( a , a ), ( b , c ), ( c , b ), ( d , d )}. This is because ( b , b ) R . Definition 3.3 A relation on a set X is called symmetric if for all x , y X , if ( x , y ) R then ( y , x ) R . The relation R in Example 3 is symmetric, but the relation in Example 2 is not. In the digraph of a symmetric relation, between any two vertices u and v , there is either no edge or a pair of opposite directed edges. Definition 3.4 A relation on a set X is called antisymmetric if for all x , y X , if ( x , y ) R and x y , then ( y , x ) R .
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SHEN’S CLASS NOTES-191 4 Example 4 (3.1.13) The relation in Example 2 (Fig. 2) is antisymmetric. The relation in Example 3 is not antisymmetric. In the digraph of a antisymmetric relation, between any two vertices u and v , there is at most one directed edge. Definition 3.5 A relation on a set X is called transitive if for all x , y , z X , if ( x , y ) and ( y , z ) R , then ( x , z ) R . Example 5 (3.1.17, 3.1.18) The relation in Example 2 (3.1.4) is transitive. You can verify every pair of ( x , y ) and ( y , z ). See the details in the book. The relation in Example 3 (3.1.7) is not transitive because ( b , c ) R and ( c , b ) R , but ( b , b ) R . Definition 3.6 A relation R on a set X is called a partial order if R is reflexive, antisymmetric, and transitive. Example 6 (3.1.20) Let X be the set of positive integers. We define a relation R by
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SHEN’S CLASS NOTES-191 5 ( x , y ) R if x divides y . Then R is a partial order. This is because: (1) ( x , x ) R because x divides itself. So, R is reflexisive.
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This note was uploaded on 04/12/2008 for the course CS 191 taught by Professor Shen during the Fall '06 term at University of Missouri-Kansas City .

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D-Math-W05-Ch3 - SHEN'S CLASS NOTES-191 Chapter Three...

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