Binary Relation A binary relation between sets A

Binary Relation: A binary relation between sets A and B is a subset of the Cartesian Product A x B. If A = B we say that the relation is a relation on A. Suppose A = {1, 3, 5, 7} and B = {2, 4, 6, 8}. Further suppose R = { (1, 2), (3, 4), (5, 6), (7, 8)}. This is a subset of A x B so is a binary relation between A and B B 8 6 4 2 1 3 5 7 Discrete Math for CS A

Binary Relation Exercise: If R = { (a, b): a < b}. Show R on the diagram below. B 8 6 4 2 1 3 5 7 Discrete Math for CS A

Binary Relations as Sets of Ordered Pairs: Because we mention one set before another in a Cartesian Product, A x B, the element, (a, b), in any relation, R, over A and B must have its first element from A and its second element from B. So we say that the elements of R form ordered pairs. Discrete Math for CS

Exercise: If X = {1, 2, 3, 4, 5, 6}, find R = { (x, y): x is a divisor of y} R = { (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6) } Discrete Math for CS

Graph Representation of a Binary Relation: If A and B are two finite sets and R is a binary relation between A and B we can represent this relation as a graph (set of vertices and edges). A = {1, 3, 5 , 7}, B = {2, 4, 6, 8} R = {(a, b): a < b} 1 2 We call this graph a directed graph. 3 4 5 6 7 8 Discrete Math for CS

Example: A = {1, 2, 3, 4, 5, 6}. R is a relation on A defined by the following directed graph. 1 2 6 Exercise: Give a rule-based definition of R 3 R = {(a, b): a divides b} 5 4 Discrete Math for CS

Matrix Representation of a Binary Relation: If A and B are finite sets and R is a binary relation between A and B then create a matrix, M, with the following properties: the rows of the matrix are indexed by the elements of A the columns of the matrix are indexed by elements of B M(ai, bj) = 1 if (ai, bj) belongs to R; 0 otherwise B A 2 4 6 8 1 0 1 1 1 3 0 0 1 1 5 0 0 0 1 7 0 0 Exercise: Give a rule-based definition of R R = {(a, b): a + 1 < b} Discrete Math for CS

Notation: If R is a binary relation on a set, X, we write x R y whenever (x, y) ε R. Example: sister_of relationship on girls in a school can be expressed as x is_sister_of y. Discrete Math for CS

Relation. Properties: Suppose R is a relation on a set A. We say R is reflexive if a R a for all a ε A. R is symmetric if x R y ==> y R x for all x, y ε A. R is antisymmetric when (x R y and y R x ==> x == y) for all x, y ε A. R is transitive when (x R y and y R z ==> x R z) for all x, y, z ε A. Discrete Math for CS

Understanding Relations as Ordered Pairs: R is reflexive if (x, x) ε R for all x ε A. R is symmetric if when (x, y) ε R then (y, x) ε R for all x, y ε A. R is antisymmetric if when (x, y) ε R and x != y then (y, x) ε R. R is transitive if when (x, y) ε R and (y, z) ε R then (x, z) ε R Discrete Math for CS

Understanding Relations as digraphs. If R is a relation represented as a di-graph then R is reflexive if every node has a loop to itself attached. R is symmetric if every directed edge is directed in both directions. R is antisymmetric if there is no bi-directional edge. If there is a directed edge from x to y and another from y to z then there is a directed edge from x to z y x z Discrete Math for CS

Understanding Relations as Matrices: Given a binary relation R on a finite set X. Let M be the matrix whose rows and columns are indexed by the elements of X. R is reflexive if the elements on the leading diagonal are all 1(T). R is symmetric if the matrix is symmetric about the main diagonal. R is antisymmetric if there are no symmetrical elements. Hence if mij == 1 the mji != 1. The text says transitivity is not readily apparent. We'll see! Discrete Math for CS