Lecture Discrete structures: Chapter 19 - Amer Rasheed

Chapter 19 - Functions. In this chapter, the following content will be discussed: Relations and functions, definition of function, examples of functions, one-to-one function, onto function, bijective function (one-to-one correspondence), inverse functions. | (CSC 102) Lecture 19 Discrete Structures Previous Lecture Summery Properties of relations Reflexive, Symmetric and Transitive Relations Equivalence Relations Properties of Congruence Modulo n Transitive closure of a relations Combining Relations Partial Order Relations Hasse Diagrams Functions Today’s Lecture Relations and Functions Definition of Function Examples of Functions One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Inverse Functions Relations If we want to describe a relationship between elements of two sets A and B, we can use ordered pairs with their first element taken from A and their second element taken from B. Since this is a relation between two sets, it is called a binary relation. Definition: Let A and B be sets. A binary relation R from A to B is a subset of A B. In other words, for a binary relation R we have R A B. We use the notation aRb to denote that (a, b) R and aRb to denote that (a, b) R. Relations If we have two sets A = {1,2,3,4,5} and B = {5,6,7,8,9} The cartesian product of A and B is A × B = { (1,5), (1,6), (1,7), (1,8), (1,9), (2,5), (2,6), (2,7), (2,8), (2,9), (3,5), (3,6), (3,7), (3,8), (3,9), (4,5), (4,6), (4,7), (4,8), (4,9), (5,5), (5,6), (5,7), (5,8), (5,9) }. The rule is to add 4: R = { (1,5), (2,6), (3,7), (4,8), (5,9) }. The domain is the set of all values which are first members of the ordered pairs in the relation, ., Dom (R) = {1, 2, 3, 4, 5} The range is the set of all values which are second members of the ordered pairs in the relation, ., Range (R) = {5, 6, 7, 8, 9} Relations 5 6 7 8 9 1 2 3 4 5 The Rule is ‘ADD 4’ One to One Relations Dom (R) = {1, 2, 3, 4, 5} Range (R) = {5, 6, 7, 8, 9} Relations Ahmed Peter Ali Jaweria Hamad Paris London Dubai New York Cyprus Has Visited Many to Many relation Dom (R) = {Ahmad, Peter, Jaweria, Hamad} Range (R) = {Paris, London, Dubai, New York, Cyprus} Note That: Ali is not in the Domain Bilal Peter Salman Ali George Aziz 62 64 66 . | (CSC 102) Lecture 19 Discrete Structures Previous Lecture Summery Properties of relations Reflexive, Symmetric and Transitive Relations Equivalence Relations Properties of Congruence Modulo n Transitive closure of a relations Combining Relations Partial Order Relations Hasse Diagrams Functions Today’s Lecture Relations and Functions Definition of Function Examples of Functions One-to-One Function Onto Function Bijective Function (One-to-One correspondence) Inverse Functions Relations If we want to describe a relationship between elements of two sets A and B, we can use ordered pairs with their first element taken from A and their second element taken from B. Since this is a relation between two sets, it is called a binary relation. Definition: Let A and B be sets. A binary relation R from A to B is a subset of A B. In other words, for a binary relation R we have R A B. We use the notation aRb to denote that (a, b) R and aRb to denote that (a, b) R. Relations If we have two sets A =

Bấm vào đây để xem trước nội dung
TÀI LIỆU MỚI ĐĂNG
283    64    2    28-03-2024
6    67    2    28-03-2024
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.