relations. ipoh kota bharu alor star seremban pasir mas perak kelantan kedah neg. sembilan...
Post on 19-Dec-2015
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Relations
Relations
Relations
Ipoh
Kota Bharu
Alor Star
Seremban
Pasir Mas
Perak
Kelantan
Kedah
Neg. Sembilan
cities states
isin
Relations
• Defining isin relation
isin == {(Ipoh,Perak), (Kota Bharu, Kelantan), (Alor Star, Kedah),
(Seremban, Neg Sembilan), (Pasir Mas, Kelantan)}
• From the above we that (Ipoh,Perak) isin• but (Ipoh, Kelantan) isin• we can deduce that, the type of isin is ℙ(cities states)
Notation for Relation
Examples
Examples
Declaring Relations
• Examples
isin : cities states• Another example
let [Author] and [Title] are given sets, then
we will have
wrote : ℙ(Author Title)
wrote : Author Title
• Representing pairs that make up a relation (x,y), we can use maplet notation x ↦y
• Using maplet notation for isin relation
{Ipoh ↦ Perak, Kota Bharu ↦Kelantan, …}
Ipoh ↦ Perak isin
{Ipoh ↦ Perak, Kota Bharu ↦Kelantan} isin
Domains and Ranges
• Domain of a relation is the set of first elements of the pairs (source) in the relation
suppose R : X Y then
dom R = { x : X | y : Y • x ↦y R}
• Range of a relation is the set of second elements of the pairs (target) in the relation
ran R = {y : Y | x : X • x ↦y R}
Example
Exercise
Assume that the definition of two relations involving the sets People, and Instruments as follows:
plays == {Ash ↦piano, William ↦ guitar, David ↦ violin,
Huw ↦ trumpet, Alice ↦ flute, Alice ↦piano, Kate ↦piano}
what are the domain and range of plays?
Restriction
• Domain restriction
getting attention to those pairs in relation whose first members are members of some other set of interest
• Example: to confine the relation wrote to those pair whose first members are in the set female --- female ⊳ wrote
• An abbreviation of either of the following
{ a: female; t : Title | a wrote t • a ↦ t }(female Title) wrote
Restriction
• Range restriction
getting attention to those pairs in relation whose second members are members of some other set of interest
• Example: restrict on second members as set of novel which is a set of titles --- wrote ▷ novel
• An abbreviation of either of the following
{ a: Author; t : Novel | a wrote t • a ↦ t }(Author Novel) wrote
Subtraction
• Domain subtraction
Getting attention to those pair in a relation whose first members are not members of some other set of interest
• Example: denote set of ordered pairs in wrote whose first members are not in female
female wrote
Subtraction
• Range subtraction
Getting attention to those pair in a relation whose second members are not members of some other set of interest
• Example: denote set of ordered pairs in wrote whose second members are not in novel
wrote novel
Inverse
Inverse
Relational Image
• Given any set of domain-type elements of a relation, we can get the subset of range elements to which they are related
Relational Composition
• Given two relations such that the range type of the first is the same as the domain type of the second
• Able to put ‘end-to-end’ to form a single relation containing all pairs joined by some common elements
• Example:
wrote : Author Title
issued-by : Title Publisher
composition published-by : wrote ; issued-by