1. Mathematical Preliminaries Hw (p.13) 1, 4, 7, 8, 9, 13, 23, 26, 30, 32

3. A set is a collection of elements SETS We write 1 is a member (or element) of set A ship is not a member (or element) of set B Membership of a given set

8. 0 2 4 6 1 3 5 7 even { even integers } = { odd integers } odd Integers

9. DeMorgan’s Laws A U B = A B U A B = A U B U

10. Empty, Null Set: = { } S U = S S = S - = S - S = U = Universal Set

11. Subset A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 } Proper Subset: A B A B U A B U

12. Disjoint Sets A = { 1, 2, 3 } B = { 5, 6} A B A B = U

14. Powersets A powerset is a set of subsets Powerset of S = the set of all the subsets of S S = { a, b, c } 2 S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } Observation: | 2 S | = 2 |S| ( 8 = 2 3 )

15. Cartesian Product A = { 2, 4 } B = { 2, 3, 5 } A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) } A X B is an ordered set, i.e. A X B ≠ B X A |A X B| = |A|·|B| Generalizes to more than two sets A X B X … X Z

17. FUNCTIONS domain 1 2 3 a b c range f : A -> B A B If A = domain then f is a total function otherwise f is a partial function f(1) = a 4 5 In general, we mean this.

19. Labeled Graph a b c d e 1 3 5 6 2 6 2

20. Walk Walk is a sequence of adjacent edges (e, d), (d, c), (c, a) is a walk from e to a of length 3 (or denoted as e-d-c-a ) Length = # of edges a b c d e

21. Path a b c d e Path is a walk where no edge is repeated Simple path : no node is repeated

22. Path a b c d e Path is a walk where no edge is repeated Simple path : no node is repeated (e, b), (b, e), (e, d), (d, c), (c, a) is a path from e to a but it is not a simple path.

23. Cycle a b c d e 1 2 3 Cycle : a walk from a node (base) to itself without repeated edges Simple cycle : only the base node is repeated Loop: an edge from a node to itself base

24. Find All Simple Paths starting from c a b c d e origin The longest simple path has at most length 4. Since every vertex can only be visited at most once, and there are 4 other vertices.

25. (c, a) (c, e) Step 1 a b c d e origin Starting from vertex c, list all outgoing edges as long as they do not lead to any vertex already used in the path. At this point, we have all paths of length one starting at c . For all vertices a , e reached by c , we list all outgoing edges originating at a or e according the same way we did before.

26. (c, a) (c, a), (a, b) (c, e) (c, e), (e, b) (c, e), (e, d) Step 2 a b c d e origin

27. Step 3 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, e) (c, e), (e, b) (c, e), (e, d)

28. Step 4 a b c d e origin (c, a) (c, a), (a, b) (c, a), (a, b), (b, e) (c, a), (a, b), (b, e), (e,d) (c, e) (c, e), (e, b) (c, e), (e, d)

29. Trees are connected directed graphs without cycles such that there is a special vertex called “root” having exactly one path to every other vertices. root leaf parent child

30. root leaf Level 0 Level 1 Level 2 Level 3 Height 3 The level associated with each vertex is the number of edges in the path form the root to the vertex. The height of the tree is the largest level number of any vertex.

31. Binary Trees A binary tree is a tree in which no parent can have more than two children. A binary tree is a tree in which no parent can have more than two children. (p.10) Example 1.5. Prove that a binary tree of height n has at most 2 n leaves.

35. Example Theorem: A binary tree of height n has at most 2 n leaves. (p.10) We want to show: L(n) ≦ 2 n for n = 0, 1, 2,…. Proof by induction: let L(i) be the maximum number of leaves of any subtree at height i

37. Induction Step From Inductive hypothesis: height k k+1 Let’s assume L(i) ≦ 2 i for all i = 0, 1, …, k need to show that L(k + 1) ≦ 2 k+1 0 … L(k) ≦ 2 k

38. L(k) ≦ 2 k L(k+1) ≦ 2 · L(k) ≦ 2· 2 k = 2 k+1 Induction Step height k k+1 (we add at most two nodes for every leaf of level k) … need to show that L(k + 1) ≦ 2 k+1 To get a binary tree of height k+1 from one of height k , we can create at most 2 leaves in place of each previous one

42. = n/m 2 m 2 = n 2 Therefore, n 2 is even n is even n = 2 k 2 m 2 = 4k 2 m 2 = 2k 2 m is even m = 2 p Thus, m and n have a common factor 2 Contradiction!

43. Languages

47. String Operations Concatenation w  = ?  w = ?

49. Reverse

59. The + Operation : the set of all possible strings from alphabet except

60. Languages Fall 2008 Automata

61. Note that: Sets Set size Set size String length

67. Reverse Hw # 10 (a) Prove or Disprove: i.e., w R  L  w  L R

69. Concatenation Hw 8. Prove

71. More Examples

74. True or False

75. True or False How to prove your answer?

76. Try Hw# 9 & 10(b) on p.28 What does w  L 2 mean? What does w  L * mean?

78. Grammars

89. Example Grammar Derivations

93. A Convenient Notation In general, we need to give a proof that a given language indeed generated by a certain grammar. Back to last Example