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Relations
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• 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
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• 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”
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• 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
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• 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?
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• 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.
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• 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.
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• 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
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• 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*
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• 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.
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• 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)
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• 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
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• 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
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• 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
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• 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
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• 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!
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• 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!
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• 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;
}}