1. Sets Model Notation Sub-Sets Operations (abstract) Operations (ADT)
Specialized ADTs based on Set
4/17/2019
CS 303 – Sets
Lecture 8

2. Sets – the Model
Set: a collection of elements (members) without repetition
{a, b, c, ... }
Usually, there is a linear order specified on the members
specified somewhere else!
not based on “position”
Well Defined: if a,b S, then either a < b, a = b, or b < a
Transitive: if a,b,c S, then (a<b) and (b<c) implies (a<c)
4/17/2019
CS 303 – Sets
Lecture 8

3. Set Notation {1, 2, ..., 1000} = {x | 0 < x <= 1000}
{1, 4} = {4, 1} <> {1, 4, 1} (which is not a set!)
Membership
x A - x is an element (member) of A
x A - x is not an element of A
Null Set
NULL
4/17/2019
CS 303 – Sets
Lecture 8

4. Sub-Sets A is a SubSet of B A is included in B B is a SuperSet of A
A is a subset of A
NULL is a subset of A
If A is a Subset of B and B is a Subset of A then A = B
if A is a Subset of B and A <> B then A is a proper subset of B
4/17/2019
CS 303 – Sets
Lecture 8

5. (abstract) Set Operations
Union: A union B = {x | (x is in A) OR (x is in B)}
Intersection: A intersect B = {x | (x is in A) AND (x is in B)}
Difference: A-B = {x | (x is in a) AND (x is NOT in B)}
[show Venn Diagrams]
4/17/2019
CS 303 – Sets
Lecture 8

6. Set ADT - Operations MakeNull(A): A = NULL
Member(x,A): b = (x is in A?)
Union(A,B,C): C = A UNION B
Intersection(A,B,C): C = A INTERSECT B
Difference(A,B,C): C = A – B
Equal(A,B): A == B?
Assign(A,B): A = B
Insert(x,A): A = A UNION {x}
Delete(x,A): A = A – {x}
Min(A): x = a | (a in A)
and for all z in A, a <= z
Merge(A,B): if A INTERSECT B is NOT NULL
then C = A UNION B,
else UNDEFINED!
Find(x): A = Ai | x is in Ai
if U = {Ai | i<>j
implies Ai INTERSECT Aj = NULL}
4/17/2019
CS 303 – Sets
Lecture 8

7. Specialized ADTs based on SET
UID – Union, Intersection, Difference
IDM – Insert, Delete, Member (Dictionary)
I(DeleteMin) – Insert, DeleteMin (Priority Queue)
...and more...much more...
4/17/2019
CS 303 – Sets
Lecture 8