1. Sets
CS3240, L. grewe

2. Goals
Describe a set at the logical level and implement a set both explicitly and implicitly

3. Sets
Set
An unordered collection of distinct values (items or components), chosen from the possible values of a single data type, called the component or base type
Base type
The type of the items in the set
Cardinality
The number of items in a set
Subset
A set contained within another set

4. Sets Universal set A set that contains all the values of the base type
Empty set
A set with no members
Set union
A set made up of all the items in either sets
Set intersection
A set made up of all the items in both sets
Set difference
A set made up of all the items in the first set that are not in the second set

5. Sets Given SetA = {X, Y, Z, A, B} SetB = {A, C, W, Z}
What is SetA union SetB?
What is SetA intersection SetB?
What is SetA - SetB (difference)?
Give a subset of SetA
Give a proper subset of SetB

6. Sets Beware At the Logical Level sets can not contain duplicates
Inserting a duplicate item into a set does not change the set
 Sets are not ordered
This does not mean that we cannot implement a set using an ordered structure with duplicates

7. Sets Explicit implementation
Each item in the base type has a representation in each instance of a set; the representation is either true (item is in the set) or false (item is not in the set)
Bit Vector 
An array of Boolean flags, with each item in the base type mapped to a specific flag (index)
Algorithms use Boolean operations
To what is space for each set instance proportional?

8. Sets
If sets A and B are represented as bit vectors, what Boolean operation would implement set union?
Set intersection?
Set difference?

9. Sets Implicit implementation (List)
The items in an instance of a set are on a list
that represents the set; those items that are not on the list are not in the set   
Algorithms use ADT List operations
To what is space for each set instance proportional?

10. Sets
If sets A and B are represented as lists, what list operations would implement set union?
Set intersection?
Set difference?
Which list implementation would be more efficient?