Slide08 Relations - Discrete Mathematics Discrete Mathematics(2009 Spring Relations(Chapter 8 5 hours Chih-Wei Yi Dept of Computer Science National

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Unformatted text preview: Discrete Mathematics Discrete Mathematics (2009 Spring) Relations (Chapter 8, 5 hours) Chih-Wei Yi Dept. of Computer Science National Chiao Tung University May 25, 2009 Discrete Mathematics Chapter 8 Relations §8.1 Relations and Their Properties Binary Relations De&nition Let A and B be any two sets. A binary relation R from A to B , written R : A $ B , is a subset of A & B . The notation aRb means ( a , b ) 2 R . If aRb , we may say ¡ a is related to b (by relation R )¢, or ¡ a relates to b (under relation R )¢. Example < : N $ N : ¡ f ( n , m ) j n < m g . a < b means ( a , b ) 2 < . A binary relation R corresponds to a predicate function P R : A & B ! f T , F g de&ned over the 2 sets A and B . Discrete Mathematics Chapter 8 Relations §8.1 Relations and Their Properties Examples of Binary Relations Let A = f , 1 , 2 g and B = f a , b g . Then R = f ( , a ) , ( , b ) , ( 1 , a ) , ( 2 , b ) g is a relation from A to B . For instance, we have 0 Ra , 0 Rb , etc.. Can we have visualized expressions of relations? Let A be the set of all cities, and let B be the set of the 50 states in the USA. De&ne the relation R by specifying that ( a , b ) belongs to R if city a is in state b . For instance, (Boulder,Colorado), (Bangor,Maine), (Ann Arbor,Michigan), (Middletown,New Jersey), (Middletown,New York), (Cupertino,California), and (Red Bank,New Jersey) are in R . ¡eats¢ : & f ( a , b ) j organism a eats food b g . Discrete Mathematics Chapter 8 Relations §8.1 Relations and Their Properties Complementary Relations De&nition Let R : A $ B be any binary relation. Then, R : A $ B , the complement of R , is the binary relation de&ned by R : & f ( a , b ) j ( a , b ) / 2 R g = ( A ¡ B ) ¢ R . Note this is just R if the universe of discourse is U = A ¡ B ; thus the name complement. The complement of R is R . Discrete Mathematics Chapter 8 Relations §8.1 Relations and Their Properties Inverse Relations De&nition Any binary relation R : A $ B has an inverse relation R & 1 : B $ A , de&ned by R & 1 : ¡ f ( b , a ) j ( a , b ) 2 R g . Examples 1 < & 1 = f ( b , a ) j a < b g = f ( b , a ) j b > a g = > . 2 If R : People ! Foods is de&ned by " aRb , a eats b ", then bR & 1 a , b is eaten by a . Discrete Mathematics Chapter 8 Relations §8.1 Relations and Their Properties Examples Example Let A = f 1 , 2 , 3 , 4 , 5 g and R : A $ A : & f ( a , b ) : a j b g . What are R and R ¡ 1 ? Solution R = & ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 1 , 4 ) , ( 1 , 5 ) , ( 2 , 2 ) , ( 2 , 4 ) , ( 3 , 3 ) , ( 4 , 4 ) , ( 5 , 5 ) ¡ Discrete Mathematics Chapter 8 Relations §8.1 Relations and Their Properties Examples Example Let A = f 1 , 2 , 3 , 4 , 5 g and R : A $ A : & f ( a , b ) : a j b g . What are R and R ¡ 1 ?...
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This note was uploaded on 10/25/2009 for the course EE 2011 taught by Professor Denny during the Spring '09 term at National Tsing Hua University, China.

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Slide08 Relations - Discrete Mathematics Discrete Mathematics(2009 Spring Relations(Chapter 8 5 hours Chih-Wei Yi Dept of Computer Science National

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