Chapter 9: Relations Relations

Transcription

1 Chapter 9: Relations Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R A B, i.e., R is a set of ordered pairs where the first element from each pair is from A and the second element is from B. If (a, b) R then we write a R b or a b (read a relates/is related to b [by R] ). If (a, b) / R, then we write a R b or a b. We can represent relations graphically or with a chart in addition to a set description. Example 1. A = {0, 1, 2}, B = {1, 2, 3}, R = {(1, 1), (2, 1), (2, 2)} Example 2. (a) Parent (b) x, y Z, x R y x 2 + y 2 = 8 (c) A = {0, 1, 2}, B = {1, 2, 3}, a R b a + b 3 (d) 1

2 Note: All functions are relations, but not all relations are functions. Definition 2. If A is a set, then a relation on A is a relation from A to A. Example 3. How many relations are there on a set with... (a) two elements? (b) n elements? (c) 14 elements? Properties of Relations Definition 3 (Reflexive). A relation R on a set A is said to be reflexive if and only if a R a for all a A. Definition 4 (Symmetric). A relation R on a set A is said to be symmetric if and only if for all a, b A. a R b = b R a Definition 5 (Anitsymmetric). A relation R on a set A is said to be antisymmetric if and only if for all a, b A. a R b and b R a = a = b 2

3 Definition 6 (Transitive). A relation R on a set A is said to be transitive if and only if for all a, b, c A. a R b and b R c = a R c Example 4. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, or transitive, where (a, b) R if and only if (a) a b (b) a b 2 Example 5. Determine whether the relation R on the set of all web pages is reflexive, symmetric, antisymmetric, or transitive, where (a, b) R if and only if (a) Everyone who has visited page a has also visited page b. (b) There are no common links found on both page a and page b 3

4 Combining Relations We can combine two relations R 1 and R 2 from A to B using the following operations (these all come from set operations that we ve discussed previously(: Intersection: R 1 R 2 = {(a, b) (a, b) R 1 and (a, b) R 2 } Union: R 1 R 2 = {(a, b) (a, b) R 1 or (a, b) R 2 } Difference: R 1 R 2 = {(a, b) (a, b) R 1 but (a, b) / R 2 } Symmetric Difference: R 1 R 2 = {(a, b) (a, b) R 1 or (a, b) R 2 but (a, b) / R 1 R 2 } Definition 7 (Composition). If R 1 is a relation from A to B and R 2 is a relation from B to C, then we define R 2 R 1 = {(a, c) a A, c C, and there exists b B such that (a, b) R 1 and (b, c) R 2 } Example 6. Let R 1 and R 2 be relations on Z where R 1 = {(a, b) a b} and R 2 = {(a, b) a is a multiple of b}. Find the following: (a) R 1 R 2 (b) R 1 R 2 (c) R 1 R 2 (d) R 2 R 1 (e) R 1 R 2 (f) R 1 R 2 (g) R 2 R 1 4

5 Definition 8. Let R be a relation on a set A. Then we define the powers of R as R 1 = R R n = R n 1 R = R } R {{ R }, for n > 1 n copies of R Example 7. A = {1, 2, 3, 4, 5}, R = {(1, 1), (1, 2), (1, 3), (2, 3), (2, 4), (3, 1), (3, 4), (3, 5)} Theorem 1. A relation R on a set A is transitive if and only if R n R for all n Z n-ary Relations Definition 9 (n-ary Relation). Let A 1, A 2,..., A n be sets. An n-ary relation on these sets is a subset R A 1 A 2 A n. The sets A 1, A 2,..., A n are called the domains of the relation and n is the degree Representing Relations In addition to charts, sets, and graphs we can represent relations using: 1. A zero-one matrix M R = [m ij ], where m ij = { 1 if (a i, b j ) R 0 if (a i, b j ) / R (This can be used for any relation from A to B.) 2. A digraph (or directed graph). Definition 10 (Digraph). A digraph consists of a set V of vertices or nodes together with a set E of ordered pairs of elements of V called edges or arcs. Given an edge (a, b), a is called the initial vertex of the edge and b is called the terminal vertex of the edge. An edge from a vertex to itself, (a, a), is call a loop. Example 8. a b c d 5

6 Example 9. Draw the divides relation on the set {2, 3, 4, 5, 6, 7, 8, 9} both as a digraph and as a 0-1 matrix. We can use digraphs and 0-1 matrices to identify whether or not a relation is reflexive, symmetric, antisymmetric, or transitive. Reflexive: Symmetric: Antisymmetric: Transitive: Theorem 2. Given relations R 1, R 2, and R on a set A with matrix representations M R1, M R2, and M R, respectively, then M R1 R 2 = M R1 M R2 M R1 R 2 = M R1 M R2 M R1 R 2 = M R2 M R1 M R n = M [n] R 6

7 Example 10. Let R 1 = {(1, 2), (2, 1), (2, 2), (3, 3)} and R 2 = {(1, 1), (1, 2), (1, 3), (3, 2)} be binary relations on the set A = {1, 2, 3}. Find M R1 and M R2 and then use them to find M R1 R 2, M R1 R 2, and M R1 R 2. Verify by computing R 1 R 2, R 1 R 2, and R 1 R 2 without matrices. Example 11. Let M R = , M S = , M P = Compute M R n, M S n, M P n, to determine if the relations R, S, and P are transitive. 7

8 9.4 - Closures of Relations Definition 11 (Reflexive Closure). Let R be a relation on a set A. The reflexive closure of R is the smallest relation containing R that is reflexive. We denote the reflexive closure by R where = {(a, a) a A}. Definition 12 (Symmetric Closure). Let R be a relation on a set A. The symmetric closure of R is the smallest relation containing R that is symmetric. We denote the reflexive closure by R R 1 where R 1 = {(b, a) (a, b) R}. Example 12. Let R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)} be a relation on the set A = {1, 2, 3, 4}. Find the reflexive closure and symmetric closure of R. Definition 13. A path from a to b in a digraph G is a sequence of edges (x 0, x 1 ), (x 1, x 2 ), (x 2, x 3 ),..., (x n 1, x n ) where n N, a = x 0, and b = x n. This path is denoted by x 0, x 1,..., x n and has length n. An empty set of edges is viewed as a path of length 0 from a to a. A path of length n 1 that begins and ends at the same vertex is called a circuit or cycle. Example 13. Let A = {a, b, c, d} and R = {(a, b), (b, a), (a, d), (d, b), (c, c), (c, b)}. Definition 14 (The Connectivity Relation). Let R be a relation on the set A. The connectivity relation R consists of all pairs (a, b) such that there is path of length at least 1 from a to b in R. Alternatively, R = i=1 R i 8

9 Example 14. Let R be the relation on the set of all people in the world that contains (a, b) iff a has met b. What is R n, where n Z +? What is R? Definition 15 (Transitive Closure). Let R be a relation on a set A. The transitive closure of R is the smallest relation containing R that is transitive. Theorem 3. The transitive closure of R is R. Example 15. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)} It turns out that if A is a set with n elements and R is a relation on A, then any time there is a path of length 1 or more from a to b in R there there is a path of length n or less from a to b in R. This means that and R = M R = n i=1 n i=1 R i M [i] R (It turns out that this is still not the most efficient way of computing R.) Example 16. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)} 9

10 Warshall s Algorithm Definition 16 (Interior Vertex). An interior vertex is any vertex in a path that is not the initial vertex or terminal vertex. (The initial or terminal vertex could be an interior vertex as long as the path visits it again without starting or ending there.) [ Theorem 4 (Warshall s Algorithm). W k = w (k) ij ], where w (k) ij = 1 if and only if there is a path from v i to v j such that all interior vertices are in the set {v 1,..., v k }. (Note: the first and last vertices in the path can be outside, length 1 paths always count since there are no interior vertices (vacuously true), and the path doesn t have to visit all of the vertices v 1,..., v k.) Example 17. R = {(1, 1), (1, 2), (2, 4), (3, 1), (4, 2)}

11 9.5 - Equivalence Relations Definition 17 (Equivalence Relation). A relation R on a set A is called an equivalence relation if and only if R is reflexive, symmetric, and transitive. If two elements a and b are related by an equivalence relation, then we write a b. 1 All elements that are related to an element a A form the equivalence class of a, denoted by [a] R (or simply [a] if there is only one relation under consideration): [a] R = {b (b, a) R} a is called a representative of the equivalence class [a] R. Theorem 5. Given an equivalence relation R on a set A and given two elements a, b A, the following are equivalent: 1. a b 2. [a] R = [b] R 3. [a] R [b] R = Example 18. (a) a b if and only if a and b have the same gender. (b) a b if and only if a and b have the same first name. 1 As was mentioned earlier in the notes, some books use the notation to denote any relation, but in this book it is only used if element are related under an equivalence relation. 11

12 (c) a b if and only if a b (mod 5). (d) [4] 5 (e) a b if and only if a and b say the same thing. (f) Definition 18 (Partition). A partition of a set S is a collection Π = {A 1, A 2,..., A m } of nonempty, pairwise disjoint subsets of S such that every element of S is in one of the subsets A i. We say that Π partitions S (or Π is a partition of S ). Theorem 6. Let R be an equivalence relation on a set S. Then the equivalence classes of R form a partition of S. Conversely, any partition Π of S defines an equivalence relation on S whose equivalence classes are the sets in Π. Equivalence relations give us a way to identify some notion of sameness (same name, same remainder, same set of a partition, etc.). 12