Predicate Calculus to Sets

1. Predicate Calculus to Sets • x : integers positive (x) is a predicate statement: • There are some integers that are “positive.” • Some “sub-group” of the domain of integers has attribute of being “positive” • Another way to specify this is via: • List the positive integers { 42, 7009, 64, 679} as a group • When we list the positive integers by unique (non-repeating) elements, then we have specified those positive integers as a SET of elements.

2. SETS • Def: A SET is a collection of objects. (In listing the collection, no duplication is allowed.) • Good example: a set called Fruit • Fruit = { apple, banana, orange, pear, grape} • Bad example ; a non-set called Fruit • Fruit = { apple, banana, orange, apple, pear} • Another example : a set called Homeowners • Homeowners = { (Joe, #203); (Sally, #7); (Tom, #143) } • note that the elements of this set are “pairs”

3. Set Membership • Def. : elements that belong to a set, S, are called members of S • If x is a member of the set, S, then we may use the following notation: x S I will use a substitute symbol, ε , in these slides. • so y ε S will mean y is a member of S. • and y ε S will mean y is not a member of S

4. Expressing with Set notation • Example from Book (Exercise 5.1 (i)) • Express the following in predicate calculus: • “All the files in the System will be read-access files or write-access files.” • \/ files: System_files ( readAccess(file) \/ writeAcess(file) ) • Express it with set notations: • file ε System_files • file ε readAccess_files \/ file ε writeAccess_files

5. Finite & Infinite sets • A set S may contain a finite number of members or an infinite number of members. • How do we “list” an infinite number of members? • A different notation is used (much like predicate calculus) : • {signature I predicate term} • Example 1 : set of integers larger than 100: • { n : N I n > 100 n} • Example 2 : set composed of pairs of integers where the 1st element is less than the 2nd: • { x,y : N I x<y (x,y) }

6. Empty Set • Def: A set that contains no member is called an empty set. • Example : consider the following set • { x : people I father (x,x) x} • There is no one who is his/her own father. • Therefore this is an empty set. • An empty set may be represented as { } or O

7. Subsets • Def: A subset of a set S is a set that contains one or more elements of S (but does not contain any element that is not a member of S). • Example : Let S = { 3, 11, 15, 4} and Z = {11, 15} • Z is a subset of S or • We can represent it with the notation Z S • A subset Z of a set S is called a proper subset of S if Z is not equal to (contains less members) S. • Example : S is the set of integers, and Z is the set of negative integers. Then Z is a proper subset of S. • Z S Note : Empty set, { }, is considered a subset of every set.

8. Power Set • Given a set S = { 1, 2, 3}, how may one increase the size of this set? • Add more members into S (we will discuss operators on sets later) • Consider permutations of subsets of S. • Def: The set of all possible subsets of S is called the power set of S. The power set of S is represented as PS or sometimes IP S. • Example ; let S = {4, 7, 2} then the PS is represented by Z, where: • Z = { s1={ }, s2={4}, s3={7}, s4={2}, s5={4,7}, s6={4,2}, s7={7,2}, s8={4,7,2} } • The power set, PS, or Z has 8 members. • So, if x ε PS , then x S. (x is a member of power set of S then it is a subset of S.)

9. Set Operations • Equality: Two sets, A and B, are equal ( = ) if they contain the same members. A = B if and only if { \/ x I (x ε A) -> (x ε B) } and { \/ y I (y ε B) -> (y ε A) } • The notion of proper subset may be expressed as: • a set A is a proper subset of set B if and only if A is a subset of B but not equal to B • A B but A = B.

10. Set Union and Intersection Operators • Union: set A union set B results in a new set C whose members are composed of members from either set A or set B. • The union operator may be specified as U. • Example : A = {5, 34, 98} and B= { 23, 34, 58} A U B = { 5, 23, 34, 58, 98} • Thus A U B = { x I (x ε A) \/ (x ε B) } • Intersection: set A intersect set B results in a new set C whose members are composed of members that are in both sets A and B. • Intersection operator may be specified as • Example : A = {5,34,98} and B { 23, 34, 58} A B = { 34 } • Thus A B = {x I (x ε A) /\ (x ε B) }

11. Set Difference • Difference : The difference of set A and set B is defined as a set C formed by removing the members which are in B from the members of A. • The difference operator is specified as \ • Example : A = { 34, 28, 5, 72} and B = { 22, 34, 5, 99} A \ B = {28, 72} • Example : “all files that are not used,” • where A is the set of all files and • B is the set of all used files. A\B = all unused files

12. Set Diagrams (Venn Diagrams) Set A Set B Union of Set A and B Set A Set B Intersection of Set A and B ( the cross section) Where is A\B in the above picture?

13. Cross Product • Cross Product: the cross product of set A and set B forms a new set C whose members are pairs made up of members from set A and B. • Cross product operator is specified with X • Example: A = { Joe, Sally} and B = { 23, 56, 89} • A X B = { (Joe,23), (Joe,56), (Joe,89), (Sally,23), (Sally,56), (Sally,89) } • A generalized form of cross product of sets A1,- - -,An would be a set composed of n-tuples or {(a1,1 , a 2,1- - - ,an,1), (a1,2 , a2,2 - - -, an,2), (a1,3, a2,3 - - - ,an,3), - - - } where a1,1, a2,1, a3,1, - - - are elements of A1, A2, A3, - - - . • Example: A = {2, 6}, B= {11, 45, 7}, and c = {90} • A X B X C = { (2,11,90), (2,45,90), (2,7,90), (6,11,90), (6,45,90), (6,7,90) }