Relational Composition

Transcription

1 Relations 4 Joseph Spring 1COM Formal Systems Relations 1 & 2 We reviewed Set concepts and considered: The concept of a relation Pairs and arrow diagram Homogeneous v Heterogeneous Source, Target, Domain and Range Total and Onto Relations 4 1 Relations 4 2 Relations 2 We continued with: Signatures Infix form Null and Full Relations The Inverse of a Relation Relations 4 3 Relations 3 We Considered: Classifying Relations» 1 - many, many - 1, 1-1, many - many Maplets Specifying Sets Database Example Company Hierarchy Set Like Things The Office Relation and its inverse The Company Switchboard Relations 4 4 This Week - Relations 4 Directed Graphs (Digraphs) Classifying Homogeneous Relations by their special properties Equivalence Relations Relations 4 5 Relations 4 6 1

2 It is often desirable to combine relations in order to form new ones. This is not a new idea. 1. We have seen this with Sets: Given sets A and B we have formed A B, A B, A\ B, A B, With predicates we have used and, or, and not Relations 4 7 The most common operation for combining relations is called We write the relational composition operator as and define it formally: R S!{ a : A ; c : C ( b : B ( a" b R) ( b" c S)) a" Relations 4 8 R S!{ a : A ; c : C ( b : B ( a" b R) ( b" c S)) a" This may be read as follows: form all pairs with a from A and c from C check each of these against the predicate - so we only keep those pairs where! (there exists) some b from B which is pointed at by a maplet in R and pointed from in a maplet in S Relations 4 R S!{ a : A ; c : C ( b : B ( a" b R) ( b" c S)) a" Question What condition needs to be satisfied in order that R and S may be composed? Relations 4 1 R S!{ a : A ; c : C ( b : B ( a" b R) ( b" c S)) a" Answer The target of R must be the source of S Can we refine this further to say ran R = dom S (No - See composition of Office and Extension) Relations 4 11 Switchboard Example The switchboard are informed that when a call comes in for an employee it is to be forwarded to any phone in that employees office To construct a record of the phone numbers on which each employee can be reached we need to compose two relations Office and Extension in that order Relations

3 We can compose the two relations since the target of the first relation Office is the source of the second relation Extension _ Office _ : Employees OfficeNumbers _ Extension _: OfficeNumbers PhoneNumbers Relations 4 13 We denote the composition of these relations as Office Extension Note that we use the relation Office first followed by the relation Extension 1. What is the signature of this new relation? 2. How do we work out what the pairs are? Relations What is the signature of this new relation? _ Office Extension _: Employees PhoneNumbers 2. How do we work out what the pairs are? Trace each possible route through Relations How do we work out what the pairs are? Office!{ John " 3, John " 5, Anne " 5, Peter " 6, Naresh " 4, Paul " 1, Darren " 6, Jim " 2, Jill " 2, Mita " 1, Mark " 8, Jane " 8} Extension!{ 1" 555, 1" 556, 2 " 5511, 2 " 5512, 3" 55, 4 " 554, 5 " 553, 6 " 552, 1 " 5515} Anne is only in office 5 which has one phone 553 so Anne " 553 Office Extension Relations 4 16 Office!{ John " 3, John " 5, Anne " 5, Peter " 6, Naresh " 4, Paul " 1, Darren " 6, Jim " 2, Jill " 2, Mita " 1, Mark " 8, Jane " 8} Extension!{ 1" 555, 1" 556, 2 " 5511, 2 " 5512, 3" 55, 4 " 554, 5 " 553, 6 " 552, 1 " 5515} Jill is in office 2 with two phones 5511 and 5512 Jill " 5511 Office Extension Jill " 5512 Office Extension Combining these gives { Jill " 5511, Jill " 5512} Office Extension Relations 4 17 Office!{ John " 3, John " 5, Anne " 5, Peter " 6, Naresh " 4, Paul " 1, Darren " 6, Jim " 2, Jill " 2, Mita " 1, Mark " 8, Jane " 8} Extension!{ 1" 555, 1" 556, 2 " 5511, 2 " 5512, 3 " 55, 4 " 554, 5 " 553, 6 " 552, 1 " 5515} Mark is in office 8 with no phone So there will be no pair involving Mark in this composition Can we refine this further to say ran R = dom S Relations

4 Can we refine this further to say ran R = dom S Answer: No! { M ark " 8} O ffice so 8 ran O ffice However 8 dom Extension so ran O ffice dom Extension Relations 4 1 ran Office = { 1, 2, 3, 4, 5, 6, 8 } dom Extension = { 1, 2, 3, 4, 5, 6, 1 } Hence ran Office dom Extension Relations 4 2 Directed Graphs (Digraphs) Used widely in Computer Science Directed Graphs (Digraphs) Relations 4 21 can be used to capture aspects of relations only suitable for homogeneous relations R: A A Basically just arrow diagrams with the source and target superimposed The circles in a digraph are called vertices Relations 4 22 Directed Graphs (Example) Consider the relation with R : A A Directed Graphs (Example) Digraph for R : A A R = { a " e, e " o, o " a, u " u, i " a, u " i } A = { a, e, i, o, u } and R = { a " e, e " o, o " a, u " u, i " a, u " i } Relations 4 23 Relations

5 Classifying Homogeneous Relations by their special properties Relations 4 25 Classifying Homogeneous Relations by their Special Properties Heterogeneous Relation we talk of a relation between sets there are two distinct sets involved Homogeneous Relation we talk of a relation on a set only one set involved Some Homogeneous relations have special properties Relations 4 26 Reflexive Property Definition A relation ~ on a set S is said to be reflexive if and only if every element of the set S is related to itself. So ~ is reflexive if and only if x ~ x for all x For the relation Same_age_as on a set of people each person will be related to his/herself For a digraph the relation is reflexive if every vertex has a loop back to itself Relations 4 27 Irreflexive Property A relation is said to be irreflexive if there is no element related to itself For a digraph a relation would be irreflexive if no vertex had a loop back to itself > is an irreflexive relation no number is greater than itself! is a reflexive relation all numbers are greater than or equal to themselves Relations 4 28 Symmetric Property A relation ~ is said to be symmetric if and only if whenever an element x is related to an element y then y is related to x We may write this as x ~ y if and only if y ~ x For a digraph a relation would be symmetric if whenever there is a line going from x to y there is also a line going from y to x Relations 4 2 Transitive Property A relation ~ is said to be transitive if and only if whenever x is related to y and y is related to z then x is also related to z We may write this as ~ is transitive if and only if whenever x ~ y and y ~ z then x ~ z For a digraph a relation would be transitive if whenever there is a line going from x to y and another going from y to z then there is a line going from x to z Relations 4 3 5

6 Transitive Property Let ~ denote the relation older_than defined on a set of people. If x is older than y and y is older than z then x is older than z So the relation older_than is a transitive relation Relations 4 31 Equivalence Relations Definition An equivalence relation ~ is a homogeneous relation that is: ~ reflexive ~ symmetric and ~ transitive Example Equality on the set of real numbers = : # # Relations 4 32 Summary Specifying Sets Database Example Directed Graphs (Digraphs) Classifying Homogeneous Relations by their special properties Equivalence Relations Relations