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