1. MCS680: Foundations Of Computer Science
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O(1)
O(n)
Running Time = 2O(1) + O(n2) = O(n2)
Algorithms
and
Analysis of
Sets
Brian Mitchell - Drexel University MCS680-FCS

2. Introduction
The set is one of the most fundamental data model in Mathematics and Computer Science
Many problems can be reduced to a set representation
Many efficient algorithms exist for processing sets
Goals
Review set definition and basic operations that can be performed on sets
Examine and analyze some set algorithms
Investigate how some “ordinary” Computer Science problems can be reduced to sets
They can then be solved using standard set algorithms
Brian Mitchell - Drexel University MCS680-FCS

3. Sets and Set Membership
When S is a set and x is a piece of quantifiable data
We need to be able to determine
Is x a member of set S
Is x not a member of set S
Generally all members of a set are related in some way
Set Notation
The expression x  S means that the element x is a member of set S
If x1, x2, ... xn are all members of the set S then we refer to set S as: S = { x1, x2, ... xn }
The empty set is denoted as 
The empty set is a set with no members
Thus for all x, x   is always false
Brian Mitchell - Drexel University MCS680-FCS

4. Definition of Sets by Abstraction
Enumerating all of the elements in a given set is not the only way we may define sets
The number of elements in the set may be large
The elements in the set may be related by some common property
Notation for abstraction: {x | x  S and P(x)}
This means that all elements x are in the set S if they satisfy property P(x)
Examples
{x | x  S and x is odd}
{A | A  T and A is a set}
{x | x  N and x is prime}
Common infinite sets
N is the set of non-negative integers, Z is the set of all integers, R is the set of real numbers, C is the set of complex numbers
Brian Mitchell - Drexel University MCS680-FCS

5. Set Operations Union, Intersection and Difference
Union: S  T
A set containing the elements in S or T, or both
Intersection: S  T
A set containing the the elements that are in both S and T
The intersection of two sets includes the elements common to both sets
Difference: S - T
A set containing those elements that are in S but not in T
The set contains the distinct elements that are in S with respect to T
Example: Let S = {1, 2, 3}, T = {3, 4, 5}
Union: S  T = {1, 2, 3, 4, 5}
Intersection: S  T = {3}
Difference: S - T = {1, 2}
Brian Mitchell - Drexel University MCS680-FCS

6. Venn Diagrams
Region 1
Region 2
Region 3
Region 4 S T
Venn diagrams are useful for visualizing set operations
Region 1 represents those elements that are in neither S or T
Region 2 represents S - T
Region 3 represents S T
Region 4 represents T - S
Regions 2, 3 and 4 combined represents S  T
Brian Mitchell - Drexel University MCS680-FCS

7. Algebraic Laws For Union, Intersection and Difference
Build up complex expressions using Union, Intersection and Difference
Example: R  (( S  T) - U)
Need to be able to determine if two expressions are equivalent
Important algebraic laws
Let the symbol  represent equivalence
Commutative Law of Union
(S  T)  (T  S)
Rationale: The element x is in S  T, if x is in X or if x is in T, or both. That is also the condition where x is in T  S.
Associative Law of Union
(S  (T  R))  ((S  T)  R)
This can be shown by the same argument used to show the commutative law
Brian Mitchell - Drexel University MCS680-FCS

8. Algebraic Laws For Union, Intersection and Difference
Commutative Law of Intersection
(S  T)  (T  S)
Rationale: The element x is in sets S  T and T  S under exactly the same circumstances.
Associative Law of Intersection
(S  (T  R))  ((S  T)  R)
Rationale: The element x must be in all three sets - S, T, and R. However it does not matter if we examine (T  R) or (S  T) first.
Distributive law of intersection over union
((T  R)  S)  ((S  T)  (R  S))
Rationale: The element x must be in S and also in at least one of T or R
Distributive law of union over intersection
((T  R)  S)  ((S  T)  (R  S))
Brian Mitchell - Drexel University MCS680-FCS

9. Algebraic Laws For Union, Intersection and Difference
Associative Law of Union and Difference
(S - (T  R) )  ((S - T) - R)
Rationale: Both sides contain an element x when x is in S but in neither T nor R.
Associative Law of Intersection
((S  T) - R )  ((S - R)  (T - R))
Rationale: An element x is in either set when it is not in R, but is in either S or T, or both.
Empty Set Identity
(S  )  S and (  S)  S
Rationale: The element x can only be in S, because it can not be in 
Idempotence: When an operator is applied to the same value, the result is the same
Union: (S  S)  S
Intersection: (S  S)  S
Brian Mitchell - Drexel University MCS680-FCS

10. The Subset Relationship
If S and T are sets, then we can say that S  T is every member of S that is also a member of T
S is a subset of T
T is a superset of S
S is contained in T
T contains S
If S and T are sets, then we can say that S  T, if S  T and there is at least one element in S that is not in T
S is a proper subset of T
T is a proper superset of S
S is a properly contained in T
T properly contains S
Brian Mitchell - Drexel University MCS680-FCS

11. The Power Set of a Set The power set of S, is the set of subsets of S
Notation: Let P(S) represent the power set of S
2S is also used
Size of a power set
A power set, P(S) has 2n members
Example:
Let S = {1,2,3}
P(S) is: {,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
|P(S)| = 8 = 23
Brian Mitchell - Drexel University MCS680-FCS

12. Cartesian Product Let A and B be two sets
The product of A and B, denoted A x B is defined as the set of pairs in which the first component is chosen from A and the second component from B
A x B = {(a,b) | a  A and b  B}
This product is known as the Cartesian Product
Example:
Let A = {1,2} and B = {a,b,c}
Thus A x B = {(1,a), (1,b), (1,c), (2,a), (2,b), (2,c)}
Size of the Cartesian Product
|A x B| = |A| * |B|
If A = {1,2} and B = {a,b,c}
Then |A| = 2, |B| = 3 and |A x B| = 6
Brian Mitchell - Drexel University MCS680-FCS

13. Binary Relations and Functions
A binary relation R is a set of pairs that is a subset of the product of two sets
If a relation R is a subset of A x B, then we say that R is from A to B
Remember that A x B  B x A
A is the domain of the relation
B is the range of the relation
Notation: Given a binary relation R on sets A and B where a  A and b  B, we can show this relation as aRb
Functions
Partial Function: Relation R has the property that for every a  A there is at most one element b B such that satisfies aRb
Total Function: Relation R has the property that for every a  A there is at exactly one element b B such that satisfies aRb
Brian Mitchell - Drexel University MCS680-FCS

14. Special Properties of Binary Relations
Transitivity
Let R be a binary relation. R is transitive whenever aRb and bRc are true, and aRc is also true
Reflexivity
Let R be a binary relation that for every element a in the domain aRa is true
Symmetric
Let R be a binary relation. R is symmetric if whenever aRb, we also have bRa
Asymmetric
Let R be a binary relation. R is asymmetric if whenever aRb, we only have bRa when a=b a a b a
never b a
Brian Mitchell - Drexel University MCS680-FCS

15. Special Properties of Binary Relations - Examples
Transitivity
Consider the relation ‘<‘ on Z, the set of integers. If a < b, and b < c, then it follows that a < c.
Consider the relation ‘’ on Z, the set of integers. If a  b, and b  c, then we can not say that a  c. This relation is not transitive
Reflexivity
The relation ‘’ is reflexive on the set of integers. For all integers a, a  a.
Symmetric
The relation ‘’ is symmetric on the set of integers. For all integers if a  b, then b  a must be true
Asymmetric
The relation ‘’ is asymmetric on the set of integers. If a  b, and b  a, then a = b
Brian Mitchell - Drexel University MCS680-FCS

16. Partial and Total Orders
Partial Order
A partial order is a transitive and antisymmetric binary relation
Total Order
A total order is a transitive, reflexive and antisymmetric binary relation where every pair of elements in the domain are comparable
If R is a total order and if a and b are any two elements in its domain, then either aRb, or bRa is true
All total orders must be reflexive because we might allow a and b to be the same element.
Example: Partial Orders and Total Orders
The relations ‘<‘ and ‘>’ are partial orders but not total orders. They are transitive and asymmetric, but not reflexive (a < a is false).
The relations ‘’ and ‘’ are total orders. They are transitive, asymmetric, and reflexive
Brian Mitchell - Drexel University MCS680-FCS

17. Closure of a Binary Relation
Closures of binary relations
Take a relation that does not have a desired property
Transitivity
Reflexive
Symmetric
Asymmetric
Add as few pairs as possible to crate a relation that does have the desired property
The resultant relation is called the closure (for the desired property) of the original relation
Closures are usually used to infer or discover missing information from the relation
Brian Mitchell - Drexel University MCS680-FCS