1. Sets

2. Sets • A set is a well-defined collection of values of the same kind (objects) • Objects can be numbers, people, letters, days, may be sets themselves • Examples

3. Sets • Small sets may be introduced by listing their elements • Other sets may be constructed using set comprehension, the power set operator, and the cartesian product • Use extension for listing elements • Examples

4. Membership of a set • We write x  s to indicate that x is an element of set s • We write (x  s ) as x  s • Examples 3  Primes 6  Primes

5. Equality of sets & Null set • Two sets of values of the same kind are equal if and only if they have the same members • Examples s == {2,2,5,5,3} t == {2,3,5} s = t • A null set is set with no members • Denoted as  or {}

6. Subsets • One set is said to be a subset of another if all the members of the one are also the members of the other • Example A is a set of all primes and B == {2,3,5} then B is a subset of A denoted by B  A

7. Union of sets • Two sets of the same kind, A and B, whose members are the members of A and B together • Denoted by A  B • Example: A == {1,2,3,4,5} B == {2,4,5,6,7} A  B == {1,2,3,4,5,6,7}

8. Intersection • Two sets of the same kind, A and B, whose members are the members that A and B have in common • Denoted by A  B • Example A == {1,2,3,4,5} B == {2,4,6,7} A  B == {2,4} • Two sets are disjoint if they have no member in common C == {6,7,8} A  C = 

9. Set Difference • Two sets A, B of the same kind, whose members are the members of A but not of B • Denoted by A\B • Example A == {1,2,3,4,5} B == {2,4} A\B == {1,3,5}

10. Set Comprehension • Defining a set by stating a property that distinguishes its member from other values of the same kind • Suppose D denotes some declarations, P denotes a predicate constraining the value, and E denotes an expression denoting a term, then {D|P•E} is called set comprehension term • Denotes a set of values consisting of all values of the term E for everything declared in D satisfying the constraint P

11. Set Comprehension • Example {x :  | x  5 • x2} denotes the same set as {0,1,4,9,16,25} • We can omit heavy dot {x :  | x  5} • We can omit constraint and constraint bar {x :  • x2}

12. Defining sets using predicates • BigCountries == {c : country | c has more than 40 million inhabitants} • MultipleOfSixes == {n : |  m :   n = m  6} • BiggestCities == {macropolis : city | co : country  macropolis is in co  ci : city  ci is in co   ci is macropolis  macropolis is bigger than ci}

13. Power Sets • The power set of a set A is the set of all its subsets. • Denoted by pA --- power set of A • Example A == {x,y} pA == {, {x}, {y}, {x,y}} • A  B has the same meaning as A  pB • Example • X == {1}, pX == {, {1}}, ppX == {, {}, {{1}}, {, {1}}} • Exercise : list the power set for {1,2,3}

14. Cartesian Product • Countries == {UK, USA, Malaysia, Iran …} • Capitals == {London, Washington, KL, Tehran, …} • CountriesAndCapitals == {(UK,London), (USA,Washington), (Malaysia,KL), (Iran,Tehran),…} • CountriesAndCapitals == {co : country; ca : city | ca is the capital of co}

15. Tuple membership • If a1…an are sets and then • Example

16. Tuple equality • if x1 = y1 and x2 = y2 … xn = yn then (x1,x2,..,xn) = (y1,y2,…,yn)

17. Component selection

18. Types

19. Sets of numbers • Integers, denoted by , is the set of positive and negative whole numbers including zero. Its type is p • Natural numbers, denoted by , is the set of whole numbers from zero onwards. The natural numbers are a subset of the integers. Its type is p • Restricted set of numbers, whose numbers lying in a certain range is called a subrange, eg. 1..4 denotes {1,2,3,4} • Finite set : 1..4 • Infinite set : , ‘the set of natural numbers that are prime’ • If X denotes some finite set, then the number of elements in the set is called its cardinality or size, and denoted by #X

20. Other forms of type • Types of ordered pair (cartesian product) involving integer  • In general if a specification has the basic types X and Y, we can have a list of other types: pX, pY, ppX, ppY, …. XX, XY, YX, YY (pX)  (pX), p(XX), (pX)  Y ….