chap mathematical preliminaries { { is positive integerxp / |pp qq pq pq - set composed of the members and set definition using set former examplethe set of all positive integers is member of set is not member of set the null or empty set cardinalitysize of set or number of members for set set is included in set qset is subset of set qset is superset of set set unionall elements appearing in or set intersectionall elements appearing in and set differenceall elements of set not in set figure set notation | (because has three membersand | (because has two membersthe union of and qwritten qis the set of elements in either or qwhich is { the intersection of and qwritten qis the set of elements that appear in both and qwhich is { the set difference of and qwritten qis the set of elements that occur in but not in qwhich is { note that and that pbut in general in this exampleq { note that the set { is indistinguishable from set pbecause sets have no concept of order likewiseset { is also indistinguishable from pbecause sets have no concept of duplicate elements the powerset of set is the set of all possible subsets for consider the set {abcthe powerset of is {{ }{ }{ }{ab}{ac}{bc}{abc}sometimes we wish to define collection of elements with no order (like set)but with duplicate-valued elements such collection is called bag to distinguish bags from setsi use square brackets [around bag' elements for the object referred to here as bag is sometimes called multilist buti reserve the term multilist for list that may contain sublists (see section