Relations We have seen several types of abstract, mathematical objects, including propositions, predicates, sets, and ordered pairs and tuples. Relations use ordered tuples to represent relationships among objects. Examples The parent relationship between specified persons can be described as a set of pairs. For instance, if Morris and Ria are the parents of Steve, then the relation would include the pairs, (M orris, Steve) and (Ria, Steve). The less-than relation on the integers consists of all pairs of integers (i,j), where i is less than j, e.g., (3,42) and (42,43). The relation that student number x is named y and majors in z is described by a set of triples, such as (124324443, M ary, CSE). Finally, the (unary) relation that k is an even integer corresponds to a subset of the integers. Relations are closely related to predicates and truth sets.
Binary Relations Let A and B are sets. A binary relation from A to B is a set R A B. In other words, a binary relation R is a set of ordered pairs (x,y) where x A and y B. If (x,y) R then we sometimes write xry. Example Take sets A = {2,6,7} and B = {1,2,5} and let R 1 be a binary relation defined by: i in A is an integer multiple of y in B. Then R 1 = {(2,1),(2,2),(6,1),(6,2),(7,1)}.
Presenting Binary Relations Binary relations have two kinds of compact visual representation, tables and graphs. The following table represents a binary relation of five pairs: R a b c 1 * * 2 * * 3 * Graphs are composed of vertices or nodes connected by edges or arcs. There is an arc from a to b if, and only if, (a,b) R. 1 A 2 B 3 C
The Parent-Of Relation The parent-of relation is a binary relation, described by pairs of names. Example Myrtle Ormonde William Joan Paula Gene Sue R G J W S M O P G[ene] J[oan] * * W[illiam] * * S[ue] M[yrtle] * O[rmonde] * * P[aula] Which representation is better for testing whether the pair (x,y) is in the relation? Which representation is better for capturing the overall structure?
Reflexive Relations A binary relation R on a set A is called reflexive if it contains all pairs (x,x) A A. Thus, if R is reflexive then the set I = {(x,x) x A} is a subset of R. A relation R on A is irreflexive if it contains no pair (x,x), i.e., the sets I and R are disjoint. In the table representation the difference between the two properties can be seen in the main diagonal of the table. R 1 2 3 R 1 2 3 1 *?? 1?? 2? *? 2?? 3?? * 3?? In a graph, the difference is in the (presence or absence of) self-loops. Is the parent-of relation reflexive, irreflexive, or neither? Is the relation reflexive, irreflexive, or neither?
Symmetric Relations A binary relation R on A is called symmetric if for all x,y A, whenever (x,y) R then (y,x) R. A relation R on A is antisymmetric if for all x,y A, if (x,y) R and (y,x) R, then x = y. The table of a symmetric relation is symmetric around the main diagonal, whereas for an antisymmetric relation there are no symmetric entries (except on the diagonal itself): R 1 2 3 R 1 2 3 1? A B 1? * 2 A? C 2 *? 3 B C? 3 *? Is the cousin-of relation symmetric, antisymmetric, or neither? Is the brother-of relation symmetric, antisymmetric, or neither?
Transitive Relations A binary relation R on A is said to be transitive if for all x, y, z in A, if xry and yrz, then xrz. If a graph represents a transitive relation and there is a path from node a to node b, then there is a single arc from a to b. Is the ancestor-of relation transitive or not? Is the friend-of relation transitive or not? The transitive closure of a relation R adds all the arcs to R necessary to make it transitive. 1 2 3 4 5 6 1 2 3 4 5 6 The ancestor relation is the transitive closure of the parent relation.
Special Binary Relations The universal relation U = A A. The empty relation R =. The identity relation I = {(x,x) x A} Which of the main properties - reflexivity, symmetry, transitivity - are satisfied by these relations?
Relational Databases Commercial database systems employ a relational model, meaning that data is stored in the form of tables of tuples, i.e., relations. A Shakespearian Killed relation would include the following entries: Killer Victim Brutus Caesar Hamlet Laertes Hamlet Polonius Laertes Hamlet Brutus Brutus Cassius Caesar Query languages like SQL can be used to extract information from a database. Example. Who killed Caesar? SELECT Killer FROM Killed WHERE victim= Caesar This reads select from relation Killed all tuples where the victim was Caesar, and report only the killer field from each. Example. Who was both a killer and a victim? (SELECT Killer FROM Killed) INTERSECT (SELECT Victim FROM Killed)
A powerful feature of relational databases is that one can combine different relations. For example, suppose there is a died-by relation: Victim Caesar Hamlet Laertes Polonius Brutus Method Daggers Sword Sword Sword Sword We can combine the two tables with a join operation, based on common fields. For example, the join of killed and died-by is: Killer Victim Method Brutus Caesar Daggers Hamlet Laertes Sword Hamlet Polonius Sword Laertes Hamlet Sword Brutus Brutus Sword Cassius Caesar Daggers Example. Which killers used daggers? SELECT Killer FROM Killed, Died-by WHERE Killed.Victim = Died-by.victim AND Method= Daggers
Equivalence Relations An equivalence relation is a binary relation that is reflexive, transitive, and symmetric. For example, the identity relation and the universal relation (on a nonempty set A) are equivalence relations. An equivalence relation defines clusters of elements of A: [x] = {y A : xry}. The sets [x] are called equivalence classes. For example, the relation on N, defined by m n iff m+n is even, is an equivalence that partitions the set of natural numbers into two subsets the sets of even and odd natural numbers, respectively.
Partitions Theorem If R is an equivalence relation on a nonempty set A, then the equivalence classes of R constitute a partition of A. Theorem If Π is a partition of A, then the relation {(x,y) : x and y belong to the same set in Π} is an equivalence relation. In other words, there is a one-to-one correspondence between equivalence relations and partitions. For example, {{a},{b},{c}} and {{a,c},{b}} are partitions of {a,b,c} and hence correspond to equivalence relations; whereas {{a},{b},{b,c}} is not a partition and does not correspond to an equivalence relation.
Congruence Modulo p Define binary relations on the integers, for all integers p greater than 1, as follows: m is congruent to n modulo p, written m n(mod p), if m n is an integer multiple of p. For example, if p = 3, Then 1 is congruent to 4 modulo p, but not congruent to 3. Let R p denote the corresponding set of pairs {(m,n) : m n(mod p)}. Are the relations R p reflexive, symmetric, and transitive?
Modular Arithmetic To visualize the relation R 12, think of grouping integers by counting around a clock. 11 0 1 10 2 9 3 8 4 7 6 5 Note that 23 = 11(mod 12). If you start at 0 and count to 23 you end up at 11. In other words, 11 and 23 have the same remainder when divided by 12. Note that if m = n(mod p) and x = y(mod p) then m+x = n+y(mod p) think about tracing out a path on the clock to prove it.
Equivalence of Strings We define a binary relation on strings as follows: Theorem Proof v w if and only if v = w. The relation is an equivalence relation. Reflexivity. We have w = w, and hence w w, for all strings w. Symmetry. If v w, then by definition v = w. By the symmetry of equality we thus have w = v, and hence w v. Transitivity. Suppose u v and v w. Then u = v and v = w and therefore, by the transitivity of equality, u = w, which implies u w. The equivalence classes for this relation are the sets Σ k of all strings of length k.
Partial Orders A binary relation on a set A is called a partial order if it is reflexive, transitive, and antisymmetric. The symbol (in infix notation) is often used to denote a partial order. We say that two elements x and y are comparable (with respect to a given partial order ) if either x y or y x. A partial order on a set A is said to be total if any two elements in A are comparable. Is the identity relation a partial order? Is the universal relation a partial order?
Partial Orders - Examples Examples of partial orders are the less-than-or-equalto relation, the divisibility relation (on the positive integers), subset relations, and ancestor relations. The following graph represents the divided-by relation on certain integers: 8 10 4 6 9 5 2 3 7 1 The antisymmetry property ensures that the graph of a partial order has no cycles (other than selfloops). If a partial order is extended to a total one we speak of topological sorting. If the set A has n elements, there are n! total orders on A.
Lexicographic Orders The words in a dictionary are arranged in a lexicographic order, which we define next. Let x and y be strings of letters. Then x L y if, and only if, 1. x is a prefix of y, or 2. x and y are equal, or 3. x can be written as uσv and y as uτw, where the letter σ alphabetically precedes the letter τ. The string u is the longest common prefix of x and y. For example, aa L ab and aa L b. The lexicographic order is also called a dictionary order. In this order there are infinite ascending chains, a L aa L aaa L Interestingly, there are also infinite descending chains, b L ab L aab L aaab L
Predecessors and Successors Let be a partial order on a set A. We say that x is a predecessor of y (and y a successor of x) if x y and x y. For example, the integer k + j is a successor of k (with respect to ) if j is positive, and a predecessor of k if j is negative. The natural number 0 has no predecessor in the set of nonnegative integers (with respect to ). We say that x is an immediate predecessor of y (and y an immediate successor of x) if there is no third element z A, such that x is a predecessor of z and z a predecessor of y. The integer k + 1 is an immediate successor of k, whereas k 1 is an immediate predecessor. Note that a rational number has no immediate predecessor or immediate successor (with respect to ).
Hasse Diagrams The Hasse diagram of a partial order on A is a directed graph with set of nodes A and an arrow from a node x to a node y whenever x is an immediate predecessor of y. These diagrams allow for a concise graphical representation of many partial orders as reflexivity and transitivity are implicitly presented. Hasse diagrams are usually drawn with their edges directed upwards (and with arrow heads left off). {a,b,c} {a,b} {a,c} {b,c} {a} {b} {c} {}
Subsets by Inclusion The graphs representing subset relations are also called hypercubes. {1, 2, 3, 4} {1, 2, 3} {1, 3, 4} {2, 3, 4} {1, 2, 4} {1, 2} {2, 3} {1, 3} {3, 4} {2, 4} {1, 4} {1} {2} {3} {4} {} Note the recursive structure of a hypercube: a d- dimensional cube is composed by connecting two (d 1)-dimensional cubes.
Infinite Hasse Diagrams Many partial orders on infinite sets can be represented by Hasse diagrams, e.g., the less-than-orequal-to relation on integers. A partial order is said to be dense if it is the case that whenever x is a predecessor of y, then there is another element z such that x is a predecessor of z and z a predecessor of y. The less-than-or-equal-to relation on the rational numbers is an example of a dense order. Dense orders cannot be represented by Hasse diagrams.
Minima and Maxima Let be a partial order on a set A. We say that x is a minimal element if it has no predecessor in A. In other words, if x is a minimal element and y x, then y = x. We say that x is a maximal element if it has no successor in A. What are the minimal and maximal elements in the divided-by relation shown previously? Does every partial order have a minimal and/or a maximal element? Is it possible for an element to be minimal and maximal at the same time?
Least and Greatest Elements If a minimal element x is a predecessor of every other element of A, then it is called the least element of A. Similarly, a maximal element that is a successor of every other element of A, is called the greatest element. For example, if we consider the subset relation on a power set P(S), then the empty set is the least element, and the set S the greatest element. The integer 0 is the least element of N (with respect to ), but there is no maximal element (and hence no greatest element either). If the set A is finite, there must be a minimal and a maximal element.