ChapterFiveSections1_2.pdf - Chapter 5 Section 5.1 1 Section Summary Relations and Functions Properties of Relations Reflexive Relations Symmetric and

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10/7/2019 1 Chapter 5 Section 5.1

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10/7/2019 2 Section Summary Relations and Functions Properties of Relations Reflexive Relations Symmetric and Antisymmetric Relations Transitive Relations Combining Relations Relations Definition: A relation R from a set A to a set B is a subset R A × B. Example : Let A = { 0 , 1,2 } and B = { a,b } {( 0, a ) , ( 0, b ) , ( 1, a ) , ( 2, b )} is a relation from A to B . We can represent relations from a set A to a set B graphically or using a table: Relations are more general than functions. A function is a relation where exactly one element of B is related to each element of A.
10/7/2019 3 Relation on a Set Definition: A relation R on a set S is a subset of S × S or a relation from S to S . Example : Suppose that S = { a,b,c }. Then R = {( a,a ) , ( a,b ) , ( a,c )} is a relation on S . Let S = { 1, 2, 3, 4 }. The ordered pairs in the relation R = {( a , b ) | a divides b } are (1,1), (1, 2), (1,3), (1, 4), (2, 2), (2, 4), (3, 3), and (4, 4). Relation on a Set ( cont. ) Question : How many relations are there on a set A ? Solution : Because a relation on A is the same thing as a subset of A A , we count the subsets of A × A . Since A × A has n 2 elements when A has n elements, and a set with m elements has 2 m subsets, there are subsets of A × A . Therefore, there are relations on a set A . 2 | | 2 A 2 | | 2 A

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10/7/2019 4 Relations on a Set ( cont .) Example : Consider these relations on the set of integers: R 1 = {( a , b ) | a b }, R 4 = {( a , b ) | a = b }, R 2 = {( a , b ) | a > b }, R 5 = {( a , b ) | a = b + 1}, R 3 = {( a , b ) | a = b or a = −b }, R 6 = {( a , b ) | a + b ≤ 3}. Which of these relations contain each of the pairs (1,1), (1, 2), (2, 1), (1, −1), and (2, 2)? Solution : Checking the conditions that define each relation, we see that the pair (1,1) is in R 1 , R 3 , R 4 , and R 6 : (1,2) is in R 1 and R 6 : (2,1) is in R 2 , R 5 , and R 6 : (1, −1) is in R 2 , R 3 , and R 6 : (2,2) is in R 1 , R 3 , and R 4 . Note that these relations are on an infinite set and each of these relations is an infinite set. Properties of Relations Let R be a relation defined on a nonempty set S. Then R is: Reflexive 𝑖? ??? ??? ?𝑙𝑙 ? ∈ ?, ?ℎ?? ?, ? ∈ ? Symmetric 𝑖??ℎ?????? ???, ?ℎ?? ??? ??? ?𝑙𝑙 ?, ? ∈ ?, ?? 𝑖? ??ℎ?? ????? 𝑖? ?, ? ∈ ?, ?ℎ?? ?, ? ∈ ? Transitive 𝑖? ?ℎ?????? ??? ??? ??? ?ℎ?? ??? ??? ?𝑙𝑙 ?, ?, ? ∈ ? ?? 𝑖? ??ℎ?? ????? 𝑖? ?, ? ∈ ? ??? ?, ? ∈ ?, ?ℎ?? (?, ?) ∈ ?