Relations

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Relations


• Definitions & Notation (1)– A binary relation from 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


• Properties of Relations– Let r be a binary relation on set S

Property Definition

reflexive x(xS (x,x) r)

symmetric x y(xS yS (x,y) r (y,x) r)

transitive x y z(xS yS zS (x,y)r (y,z)r (x,z)r)

antisymmetric x y(xS yS (x,y) r (y,x) r x = y)

Don’t confuse antisymmetric with “not symmetric”!

Likewise irreflexive and “not reflexive”


• Relations built from Relations (1)– Closure

• Definition: A binary relation r* on a set S is the closure 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


• 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 a has met b.• 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!

What is the difference?


• 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.


• 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.


• 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…

rbaif

rbaifm

ji

jiij ),(0

),(1


• 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}

kijk

r

wW

MW

0

else

ifwk

ij 0

1

Note: Wn = Mr*


• Application: Warshall’s Algorithm (2)– Example

0100

1001

0101

1000

0W

0100

1001

1101

1000

1W 2W

1101

1001

1101

1000

3W

1101

1101

1101

1101

4W

v 1 v 2

v 3v 4

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 throughonly 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 throughonly v1, v2, or v3.

W4 has a 1 as its (i,j)th entry if there is a sequence from vi to vj moving throughonly v1, v2, v3, or v4.


• 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

– AlgorithmW = Mr

for k = 1 to n for i = 1 to n for j = 1 to n

wij = wij (wik wkj)


• 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


• 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


• 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 meetx.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!

irreflexive

antisymmetric

transitive


• 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

dataisEqual = (this.x ==

castedThat.x);}return isEqual;

}}

ReflexiveSymmetricTransitive


• 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

dataisEqual = (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!


• 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!


• 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 databoolean 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;

}}

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