1. Relations

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

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

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 female wrote

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 wrote novel

17. Inverse

19. 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

21. Relational Composition

22. 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