Relations

1. Relations

2. Definitions & Notation (1) • A binary relationfrom A to B is a subset of A x B • A binary relation on A is a subset of A x A • A binary relation is defined by • Enumerating elements • Relations definition: x r y  x + y is odd • Binary relations can be • one-to-one • one-to-many • many-to-one • many-to-many • An n-ary relation on S1,S2,…,Sn is a subset of S1 x S2 x S3 x … x Sn • Si are called the domains • n is called the degree

3. Properties of Relations • Let r be a binary relation on set S Don’t confuse antisymmetric with “not symmetric”! Likewise irreflexive and “not reflexive”

4. Relations built from Relations (1) • Closure • Definition: A binary relation r* on a set S is theclosure of a relation r on S with respect to property p if • r* has property p • r  r* • r* is a subset of every other relation on S that includes r and has property p. • Composite • Definition: Let r be a relation from A to B and s be a relation from B to C. The composite (r  s) is the relation consisting of ordered pairs (a,c) where a A, c C, and for which there exists an element b  B such that (a,b)  r and (b,c)  s. • Definition: Let r be a relation on set A. The powers rn, n = 1,2, … are define recursively by • r1 = r • ri+1 = ri  r

5. What is the difference? • Relations built from Relations (2) • Example: Let r be the relation on the set of all people in the world that contains (a,b) if ahas metb. • What is rn? • Those pairs (a,b) such that there are people x1,x2,…,xn-1 such that a has met x1, x1 has met x2, …, and xn-1 has met b. • What is r*? • Those pairs (a,b) such that there is a sequence of people, starting with a and ending with b, in which each person in the sequence has met the next person in the sequence. Who cares? Wait for graphs!

6. Types of Relations (1) • Partial Ordering • Definition: A relation r on a set S is a partial ordering if it is reflexive, antisymmetric, and transitive. • (S, r) is called a partially ordered set or poset • The elements a and b of a poset (S, r) are called comparable if either (a,b)  r or (b,a)  r • Strict Partial Ordering • Definition: A relation r on a set S is a strict partial ordering if it is irreflexive, antisymmetric, and transitive. • Total Ordering • Definition: A relation r on a set S is a total ordering if it is (S,r) is a poset and every two elements of S are comparable. • Strict Total Ordering • Definition: A relation r on a set S is a strict total ordering if it is (S,r) is a strict poset and every two elements of S are comparable.

7. Types of Relations (2) • Equivalence Relations • Definition: A relation r on a set S is an equivalence relation if it is reflexive, symmetric, and transitive. • Two elements related by an equivalence relation are said to be equivalent • The set of all elements that are related to an element a of S is called the equivalence class of a. • A partition of a set is a collection of disjoint nonempty subsets of S such that they have S as their union. The equivalence classes of r form a partition of S.

8. Application: Relation Representation • Enumeration • list the ordered pairs • Zero-One Matrix • Suppose r is a relation from A {a1,a2,…,am} to B {b1,b2,…,bn} • r can be represented by matrix Mr = [mij] where • Digraph • A relation r on a set S is represented by a directed graph (digraph) that has the elements of S as it vertices and the ordered pairs (a,b) where (a,b)  r, as edges. • So how do we represent digraphs in a computer? Later…

9. Application: Warshall’s Algorithm (1) • Stephen Warshall circa 1960 • Algorithm to find the transitive closure of a set S • transitive closures are particularly interesting in that they provide “connection” information • Suppose r is a relation on S with n elements • Let a1, a2, …, an be an arbitrary listing of those elements • If a,x1,x2,…,xm-1,b is a sequence in the transitive closure, then the xis are called the interior elements of the sequence. • Warshall’s algorithm is based on the construction of a series of zero-one matrices (W0,W1, …, Wn) where • where • there is a sequence from xi to xj using only interior elements {x1,…,xk} Note: Wn = Mr*

10. Application: Warshall’s Algorithm (2) • Example W0 is the matrix of the relation. W1 has a 1 as its (i,j)th entry if there is a sequence from vi to vj moving through only v1. Since no edges go into v2, W2 is the same as W1. W3 has a 1 as its (i,j)th entry if there is a sequence from vi to vj moving through only v1, v2, or v3. W4 has a 1 as its (i,j)th entry if there is a sequence from vi to vj moving through only v1, v2, v3, or v4.

11. Application: Warshall’s Algorithm (3) • How do we calculate the Wis? • We can compute Wk directly from Wk-1 • Adding vk to Wk-1 can do one of two things: • Leave a sequence untouched (can’t use vk) • Wk at (i,j) is 1 only if Wk-1 at (i,j) is a 1 • Add a sequence from vi to vk to vj • Wk at (i,j) is 1 only if Wk-1 at (i,k) is 1 and Wk-1 at (k,j) is 1 • Algorithm • W = Mr • for k = 1 to n • for i = 1 to n • for j = 1 to n • wij = wij  (wik  wkj)

12. Application: Relational Databases (1) • Recall from CS 185 • E-R Modeling • Attributes • One-to-One, One-to-many, Many-to-one, and Many-to-Many • Both “Entity Sets” and “Relations” in Databases are relations in the mathematical sense • Table is a set of n-tuples (rows) • No duplicates and No order • a table is a subset of D1 x D2 x … x Dn where Di is the domain from which attribute Ai takes its value • therefore a table is an n-ary relation on Dis • E-R Relations have Di in one table the same as Di for the primary key of another • Joins the attributes into a new cross-product • therefore a relation is an m-ary relation on Dis

13. Application: Relational Databases (2) • Operations on Relations • restrict • Let r be an n-ary relation and c a condition that elements of r must satisfy. Then the restrict operator rc maps the n-ary relation r to the n-ary relation of all n-tuples from r that satisfy the condition c. • leads to the SQL “where” clause • project • The projection Pi1,i2,…,im maps the n-tuple (a1,a2, …,an) to the m-tuple (ai1,ai2,…,aim) where m n. • leads to the SQL “select” clause • join • Let r be a relation of degree m and s a relation of degree n. The join jp(r,s), where p m and p  n, is a relation of degree m + n – p that consists of all (m + n – p)-tuples (a1,a2,…,am-p,c1,c2,…,cp,b1,b2,…,bn-p) where the m-tuple (a1,a2,..,am-p,c1,c2,…,cp)  r and the n-tuple (c1,c2,…,cp,b1,b2,…,bn-p)  s. • leads to the SQL “from a,b,…,c” clause

14. Application: compareTo in JCF • Java Collections Framework provides a collection of container classes • Example: HashMap, HashSet, … • Some collections are ordered • Example: TreeSet, … • How does Java order the items in the collection? • By use of the compareTo(Object obj) method • By definition, compareTo(Object obj) must define a strict total ordering of all elements in the container • compareTo(Object obj) must meet • x.compareTo(y) == -1 * y.compareTo(x) • x.compareTo(y) == y.compareTo(z) == x.compareTo(z) • x.equals(y)  x.compareTo(y) == 0 • failure to meet these requirements will result in unexpected behavior • for example, Sets with duplicate objects! antisymmetric transitive irreflexive

15. Application: equals in Java (1) • According to Java API “The equals method implements an equivalence relation on non-null object references” • Therefore a.equals(b) must behave the same as b.equals(a) • Most implementations fail on this property (1) • class A { • private int x; • public boolean equals(Object that) { boolean isEqual = false; if ((that != null) && (that instanceof A)) { A castedThat = (A) that; // perform comparisons on private data isEqual = (this.x == castedThat.x); } return isEqual; } • } Reflexive Symmetric Transitive

16. Application: equals in Java (2) • Most implementations fail on this property (2) • class B extends A { • private int y; • public boolean equals(Object that) { boolean isEqual = false; if ((that != null) && (that instanceof B)) { B castedThat = (B) that; // perform comparisons on private data isEqual = (this.y == castedThat.y); } return (isEqual && super.equals(that)); } • } instanceA.equals(instanceB) would return true, but instanceB.equals(instanceA) would fail the instanceof test and return false!

17. Application: equals in Java (3) • Correct Definition (1) • abstract class T { public final boolean equals(Object that) { boolean isEqual = false; if ((that != null) && (that instanceof T)) { T castedThat = (T) that; if (this.getTypeEquiv().equals( castedThat.getTypeEquiv())) { isEqual = localEquals(that); } } return isEqual; } protected boolean localEquals(Object that) { return true; // to stop the chaining }abstract protected Class getTypeEquiv(); • } Top of hierarchy!

18. Application: equals in Java (4) • Correct Definition (2) • class A extends T {private int x; • protected boolean localEquals(Object that) { A castedThat = (A) that;// perform comparisons on private data boolean isEqual = (this.x == castedThat.x); return (isEqual && super.localEquals(that)); } protected Class getTypeEquiv() { Class result = null; try { // will never fail, but must try/catch result = Class.forName(“A”); } catch (ClassNotFoundExeception e) { } return result; } • }