离散数学英文课件：DM_lecture8 Relations.ppt

§8.1 Relations Mathematical objects designed to specify and describe relationships between elements of a set or sets. A function f from a set A to a set B assigns exactly one element of B to each element of A. In relations there is no restriction. An element in A can be assigned more than one element in B. Relations are generalization of functions; they can be used to express a much wider class of relationships between sets. Binary Relations Let A, B be any sets. The relationship between the elements of these two sets are represented by a Binary Relation. A binary relation from A to B is a set R where a R b denotes (a, b) ∈ R A X B. a is said to be related to b by R. The elements of A X B and R are said to be ordered pair. Let A = {2, 3, 4}, and B ={4, 5}. Then A X B ={(2, 4), (2, 5), (3, 4), (3, 5), (4, 4), (4, 5)}. Binary Relations For finite sets A, B with |A| = m and |B| = n, there are 2mn relations from A to B, including the empty relation as well as the relation A X B. E.g., let A = {2, 3, 4}, and B ={4, 5}. Then relations from A to B: ? {(2, 4)},… {(2, 4), (2, 5)},… {(2, 4), (3, 4), (4, 4)},… A X B Relations as graph Let A = { a, b, c}, B = {1, 2, 3, 4}, R is defined by the ordered pairs or edges {(a, 1), (a, 2), (c, 4)} can be represented by the digraph D: Properties of Relations Of special Binary Relation is the relation of the set A on A i.e. A X A called a binary relation on A. The Relation is called Reflexive, which simply means that each element a of A is related to itself. R on a set A is called reflexive if ?a∈ A ((a, a) ∈ R) where A is the universe of discourse. Properties of Relations Let A = {1, 2, 3, 4} R1 and R2 are examples of reflexive relations since they contain all pairs of the form (a, a). R1 = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (3, 3), (4, 1), (4, 4)}. R2 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)}. Properties of Relations Relation R on set A is called symmetric if