CSE 1400 Applied Discrete Mathematics Relations Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Relations and Their Graphs 1 A Relation’s Domain, Co-domain, and Range 2 A Sampling of Relations 3 Equality 3 Less than 3 Divides 3 Congruence Modulo n 4 Perpendicular on Lines 5 The Incestuous and Empty Relations 5 A Relation is a Set of Ordered Pairs 6 The Inverse Relation 6 Counting Relations 7 Relational Properties 7 Reflexive Property 7 Symmetric Property 8 Antisymmetric Property 10 Transitive Property 11 Orders and Equivalences 11 Orders 12 Well-Ordered Sets 13 Equivalences 13 Equivalence Relations Partition a Set 14 Stirling Numbers of the Second Kind 16 cse 1400 applied discrete mathematics relations 2 Problems on Relations 18 Abstract A relation ∼ describes how things are connected. That a thing a is related to a thing b can be represented by 1. An ordered pair (a, b). • 2. An directed edge a •b . 3. Or more commonly, simply using relational notation a ∼ b. A relation is 1. A set G of ordered pairs. 2. A directed graph G of nodes and edges. 3. A matrix of True and False values. Higher-dimensional relations among a, b, c or more parameters can be defined. Higher-dimensional relations occur as tables in relational databases and as data in multi-variable problems. Relations and Their Graphs A relation is a set of ordered pairs. A relation can be pictured of as a graph G. •g G = f(x, y) : x ∼ yg = f(x, y) : x is related to y.g h• There are many examples of relations. You are, no doubt, familiar d• c• with relations among people: Mother-Daughter, Father-Son, Parent- Child, Aunt-Nephew. Familial relations often become murky. We study relations that can be precisely defined. A few common rela- •e • f tions are equality, congruence mod n, less than, divides, subset, and •a perpendicular. •b This In this course, relationships will be between two things a and b. graph represents the relation Relationships among 3 or more things are common and useful, but G = f(a, b), (b, c), (c, d), (d, h), (h, f ), ( f , e), (e, a)g these ideas are not within the scope of this course. The course studies binary relations. A Relation’s Domain, Co-domain, and Range The things involved in a relation need names:Call them the things x and y. Write x ∼ y to express the phrase “x is related to y.” cse 1400 applied discrete mathematics relations 3 The value x belongs to a set X called the domain of ∼. The value y belongs to a set Y called the co-domain of ∼. The domain X is the set of elements that appear on the left-hand side of ∼. For this course, you can assume that every element in X appears on the left-hand side of ∼, that is, every relation we en- counter is total, defined on all members of X. A relation ∼ is said to be total when The co-domain Y is the set of elements that can appear on the every x 2 X occurs on on the left-hand side of ∼ for some y 2 Y. right-hand side of ∼. A potentially smaller set is the range of ∼. (8x 2 X)(9y 2 Y)(x ∼ y) The range is the set of elements in Y that actually do appear on the right-hand side of ∼. In general, not every element y in Y occurs If there is an x 2 X such that no y 2 Y is related to x, then ∼ is said to be on the right-hand side of ∼. A relation is into its co-domain and partial. onto its range. That is, the co-domain of a relation, is the set of (9x 2 X)(8y 2 Y)(x 6∼ y) values y that could be related to some x. When a relation is onto its An instance of a partial relation on the co-domain every element in Y is related to some element x in X, real numbers is the pairing (x, 1/x), written which is undefined (has a pole) at x = 0. (8y 2 Y)(9x 2 X)(x ∼ y) A relation maps an element x in the domain X onto zero or more elements y in the range A ⊆ Y. A relation maps its domain into its Consider the graph in figure 1 that depicts a relation from vertices co-domain and onto its range. x0, x1, x2, x3 in domain X to vertices y0, y1, y2, y3 in co-domain Y X Y Figure 1:Domain X = fx0, x1, x2, x3g, Co-Domain Y = fy , y , y , y g, x3 y3 0 1 2 3 Range A = fy0, y1, y3g, x2 y2 x1 y1 x0 y0 Here are some things to notice about figure 1. 1. The relation is total: there is at least one directed edge from each vertex in X. 2. The relation is not onto its co-domain. 3. The relation is non-deterministic: A given input x can be related to many outputs y. For instance, in figure 1 x0 is related to y0 and y1. A relation is partial when a given input x cannot be related to any output y. cse 1400 applied discrete mathematics relations 4 X Y Figure 2: A partial relation: The relation is not defined on x . x3 y3 1 x2 y2 x1 y1 x0 y0 A Sampling of Relations You are familiar with many mathematical relations: Equality, less than, multiple of, and so on. These relations are between two things : a and b, and are called binary relations. Multidimensional relations are common in practice. Equality Equality is the most basic relation. 25 −10 72 5 = = = 5 −2 72 5 equal :: Integer -> Integer -> Bool equal a b = (0 == b - a) The name (5, 25/5, . .) can change, the thing remains the same. Figure 3: Haskell for deciding equality on the integers. Less than lessthan :: Integer -> Integer -> Bool Less than establishes an order on the integers. lessthan a b = (0 < b - a) Figure 4: Haskell for deciding less than ··· < −3 < −2 < −1 < 0 < 1 < 2 < 3 ··· on the integers. Divides 7 divides 35. “b divides a” is written b j a A natural number b divides a natural number a when “b does not divide a” is written there is an quotient q 2 N such a = bq. Divides is a rela- tion on N × N. b - a For instance, 15 divides 60 because “a is a multiple of b” is written a ≡ 0 mod b 60 = 15 · 4 “a is not a multiple of b” is written Stated the other way around, 60 is a multiple of 15. On the other a 6≡ 0 mod b hand, 60 is not a multiple of 8; 8 does not divide 60. 60 divided by 8 leaves a remainder of 4. In general, let a and b be natural numbers. Then b divides a, if there is a natural number q such that a = bq. If b divides a, then a is a multiple of b. cse 1400 applied discrete mathematics relations 5 The natural numbers can be partially ordered by the divides rela- 7 tion. The Hesse graph in figure 5 illustrates this ordering for the first 3 6 9 few natural numbers. Given natural numbers a and b, it is straightforward to write code 1 2 4 8 0 to test if a divides b. 5 10 Congruence Modulo n 11 Figure 5: 1 is the smallest natural number with respect to divides; 1 divides every natural number. 0 is the Congruence mod n has applications in many areas, cryp- largest natural number with respect to tology is just one interesting application that can be divides; every natural number divides 0. named. When natural number n divides the difference a − b, the integers a and b are said to be congruent mod n. For instance, the following pairs are congruent mod 2. (15, 9), (26, 30), (−9, 21), (6, −28), (17, −17) divides :: Nat -> Nat -> Bool while the following pairs are congruent mod 3. divides a b = (a ’div’ b == 0) ( ) ( ) (− ) ( − ) ( − ) Figure 6: Haskell for deciding divides 15, 9 , 26, 29 , 8, 19 , 6, 27 , 17, 16 on the natural numbers. n is called the modulus. Congruence collects pairs of integers based on their remainder when divided by n. Mod 2 collects pairs (a, b) that are both even (2s, 2t) or both odd (2s + 1, 2t + 1). Congruence mod 2 partitions the integers into two If both a and b are even, then their equivalence classes: The even integers and the odd integers. difference is divisible by 2. If both a and b are odd, then their difference is Mod 3 collects pairs (a, b) based on remainders upon division by divisible by 2. 3. There are 3 cases. 1. Both a and b are multiples of three: (3s, 3t) 2. Both have a remainder of 1 when divided by three: (3s + 1, 3t + 1) 3. Both have a remainder of 2 when divided by three: (3s + 2, 3t + 2) Congruence mod 3 partitions the integers into three equivalence classes: Integers that are multiples of 3, integers that are multiples of 3 plus 1, and integers that are multiples of 3 plus 2. In general, two integers a and b are congruent mod n if a − b is a multiple of n. Congruence mod n partitions the integers into n equivalence classes. A convenient notation for these sets of integers is [0], [1], [2],..., [n − 1] where syntatic sugar can be added to denote [r] = fnk + r : k 2 Zg the modulus: [r]n.