Relations

1. Relations

2. Relations

3. Relations cities states isin Ipoh Kota Bharu Alor Star Seremban Pasir Mas Perak Kelantan Kedah Neg. Sembilan

4. 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)

5. Notation for Relation

6. Examples

7. Examples

8. 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

9. 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

10. 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}

11. Example

12. 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?

13. 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

14. 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

15. 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

16. 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