Relations. Ipoh Kota Bharu Alor Star Seremban Pasir Mas Perak Kelantan Kedah Neg. Sembilan...
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Transcript of Relations. Ipoh Kota Bharu Alor Star Seremban Pasir Mas Perak Kelantan Kedah Neg. Sembilan...
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