1. Sets

2. Introduction • Sets are an important mathematical data structure • Naturally occurring • Theoretically described • Powerful notation • Definition • An unordered collection of distinct objects that share one or more common property • Unordered means no inherent order • Distinct means there are no duplicate member • x[(x  S  P(x))  (P(x)  x  S)] Element of

3. Notation and Terms • Sets are denoted with capital letters • Curley brackets are also used {a,b,c} • Empty Set denoted by  or {} • not the same as {} • Finite sets can be defined by enumerating the elements of the set • Infinite sets require definition of defining property • { x | P(x) } • Example: { x | x  N x  10 } Such that set of all nonnegative integers

4. Notation and Terms • Other Commonly Used Sets • Z : set of all integers • Q: set of all rational numbers • R: set of all real numbers • C: set of all complex numbers • Z+: set of all positive integers • Subset (A  B) • x (x  A  x  B) • A  B proper subset • x (x  A  x  B)  x(x  B  x A) • Equal Sets (A = B) • x [(x  A  x  B)  (x  B  x  A)] • (A  B)  (B  A)

5. Notation and Terms • Power Set ( (S) ) • | (S)| = _________ where n = |S| • Note: the empty set is a subset of every set • Venn Diagrams

6. Set Operations • Union (A  B) • {x | x  A  x  B} • Intersection (A  B) • {x | x  A  x  B} • Complement (A) • For a set A (S), A is {x | x  S  x A} • Set Difference (A – B) • { x | x  A  x B } • Cartesian Product (A  B) • {(x,y) | x  A  y  B} Ordered pair

7. Set Operations What can you conclude about the sets A & B if… A – B = B – A A  B = A A  S =  A  B = A (A  B) = B

8. Set Identities

9. Practice Problems Mathematical Structures for Computer Science Pg. 202 # 10, 12, 18, 19, 20, 23, 30, 38, 39, 40, 50, 51, 55, 58, 62, 66, 69

10. Application: Minimal Spanning Trees • A tree is nothing more than an undirected graph without cycles • A spanning tree is a tree that contains all the vertices in the graph • There can be many such spanning trees and we usually want to find the optimal (minimal) weighted one • Thus, this is an optimization problem

11. Application: Minimal Spanning Trees 1 1 1 A B A B A B 3 3 2 2 6 6 C 4 D C D C 4 D 2 5 5 2 E E E Graph ST MST

12. Application: Minimal Spanning Trees • PRIM’s Algorithm • Given a set of vertices, V, and a set of edges, E • Start with an empty set of edges, F, and a set of vertices, Y, initialized to contain an arbitrary vertex, say ‘A’ • At each step find the minimum weight edge, where one end is from (V-Y) and the other from Y • Add this edge to F • Add the vertex from the edge not in Y to the set Y • The problem is solved when Y = V

13. Application: Minimal Spanning Trees 1 1 1 A B A B A B 3 3 3 2 2 2 6 6 6 C 4 D C 4 D C 4 D 2 5 2 5 2 5 E E E 1 1 A B A B 3 3 2 2 6 6 C 4 D C 4 D 2 5 2 5 E E

14. Application: Minimal Spanning Trees • KRUSKAL’s Algorithm • Given a set of vertices, V, and a set of edges, E • Start by creating disjoint subsets of V, one for each vertex, containing only that vertex • Look at each edge in turn, from minimal weight to maximum weight • If the edge connects two vertices in disjoint subsets then the edge is added to the MST and the two disjoint subsets are merged • Otherwise we throw away the edge and leave the subsets alone • Stop when we only have 1 disjoint subset or we run out of edges to consider

15. Application: Minimal Spanning Trees 1 A B 3 2 6 1 1 A B A B A B C 4 D 2 5 E C D C D C D 2 E E E 1 1 1 A B A B A B 3 2 2 2 C D C 4 D C D 2 2 2 E E E

16. Application: Minimal Spanning Trees A B 2 6 3 1 1 3 E F 1 2 5 3 C D 4

17. Greedy Algorithms • Arrives at a solution by making a sequence of choices, each of which looks the best at the moment • Hope that globally optimal solution will be obtained from locally optimal choices • Prim’s & Kruskal’s Algorithms • Always pick the minimal weight edge to add to the set of edges • Ncoins problem • Always pick the largest coin available • Does this always work?

18. Greedy Algorithms • Not every greedy algorithm produces optimal solution • Must prove that a greedy algorithm produces optimal solution to a problem in order to use it • Proof for Prim’s algorithm see pg 149.

19. Greedy Algorithms • Basic approach to greedy algorithms: • Start with { } • Add items to the set in sequence as follows: • Selection: choose the next item according to a greedy criterion • Feasibility: determine if the new set is feasible – can it lead to a solution? • Solution: is the new set a solution? • Stop when the set represents a solution

20. Greedy Algorithms – more practice Foundations of Algorithms Chapter 4 Practice Problems: Pg. 181 # 1, 2, 5, 6, 9, 43, 44

21. Application: Set Representation • How can we effectively represent sets in a computer? • Solution I : Store as a collection of distinct items • Tends to be inefficient since set operations will then require massive amounts of searching • Solution II: Store in some arbitrary but well defined order • Let U be the universal set (assume finite & reasonable size) • Order U, for instance, a1, a2, …, a3 • Represent A  U as a bit string of length n where the ith bit in the string is 1 if ai  A and 0 if ai  A. • Intersection reduces to  • Union reduces to  • Compliment reduces to  • Inefficient in terms of subset repetition