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, e , in these
  slides.
• so y e S will mean y is a member of S.
• and y e 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 e System_files
• file e readAccess_files \/ file e
  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 gt 100 n
• Example 2 set composed of pairs of integers
  where the 1st element is less than the 2nd
• x,y N I xlty (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 , s24, s37, s42, s54,7,
  s64,2, s77,2, s84,7,2
• The power set, PS, or Z has 8 members.
• So, if x e 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 e A) -gt (x e B) and
  \/ y I (y e B) -gt (y e 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 e A) \/ (x e 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 e A) /\ (x e 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)