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D-Math-W05-Ch3

# 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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