Relations

Ipoh Kota Bharu Alor Star Seremban Pasir Mas Perak Kelantan Kedah Neg. Sembilan citiesstates 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

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

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