Relations

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# Relations - PowerPoint PPT Presentation

Relations. Definitions &amp; 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

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Presentation Transcript

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

Properties of Relations

• Let r be a binary relation on set S

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 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

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!

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…

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*

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.

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)

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 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

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

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!

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 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; }
• }