Mathematics Fundamentals and Applications in Computer Science

1. CSCI 115 Course Review

2. Chapter 1 – Fundamentals1.1 Sets and Subsets • Set equality • Special sets (Z, Z+, Q, R, {}) • Power sets • Cardinality • Subset notation and meaning

3. Chapter 1 – Fundamentals1.2 Operations on Sets • Union • Intersection • Complement • Symmetric Difference • Addition Principles • For 2 sets: |A  B| = |A| + |B| - |A  B| • For 3 sets: |A  B  C| = |A| + |B| + |C| - |A  B| - |B  C| - |A  C| + |A  B  C|

4. Chapter 1 – Fundamentals1.3 Sequences • Definition • Characteristic Function (and computer representations) • Countable and Uncountable Sets • Regular Expressions

5. Chapter 1 – Fundamentals1.4 Division in the Integers • Prime numbers • Divides (a | b) • GCD • LCM • Number bases • Cryptology – Sir Francis Bacon’s code

6. Chapter 1 – Fundamentals1.5 Matrices • Terminology • Operations (add, sub, multiply) • Boolean Matrices and Operations • Join (or) • Meet (and) • Boolean Product

7. Chapter 1 – Fundamentals1.6 Mathematical Structures • Structure • Objects • Operations • Possible existence of identity • Other properties (Associative, commutative, etc.)

8. Chapter 2 – Logic2.1 Propositions and Log Ops • Statements • Logical operators (and, or, not) • Truth Tables • Quantifiers • Universal • Existential

9. Chapter 2 – Logic2.2 Conditional Statements • Conditional • Biconditional • Converse • Inverse • Contrapositive • Standard Truth Tables

10. Chapter 2 – Logic2.3 Methods of Proof2.4 Mathematical Induction • Direct Proof • Contradiction • Other tips / techniques • (even / odd, etc.) • Mathematical Induction

11. Chapter 3 – Counting3.1 Permutations and 3.2 Combinations • Principle of Counting • Permutations: • Ex: How many ways to seat 7 people • Combinations: • Ex: How many 7 card hands can be dealt from 52 card deck

12. Chapter 3 – Counting3.4 Elements of Probability • Sample Spaces and Events • Probability spaces • Equally likely outcomes • Expected values

13. Chapter 3 – Counting3.5 Recurrence Relations • Techniques • ‘Eyeball’ • Backtracking • Linear Homogeniety

14. Chapter 4 – Relations and Digraphs4.1 Product Sets and Partitions • Product Sets • Ex. R x R • Partitions

15. Chapter 4 – Relations and Digraphs4.2 Relations and Digraphs • Relations – What are they? • Domains • Ranges • Relation • Element • Subset • Representations • Ordered Pairs • Matrix • Digraph • Restriction to a subset

16. Chapter 4 – Relations and Digraphs 4.3 Paths in Relations and Digraphs • Paths • Compositions • Relations

17. Chapter 4 – Relations and Digraphs4.4 Properties of Relations • Reflexive • Irreflexive • Symmetric • Asymmetric • Antisymmetric • Transitive

18. Chapter 4 – Relations and Digraphs4.5 Equivalence Relations • Equivalence Relation: Ref, Symm, Trans • Ex: R on Z+ by aRb iff a=b(mod 2) • Equivalence Classes • A/R (Partition)

19. Chapter 4 – Relations and Digraphs4.6 Computer Representations • Linked Lists • Different implementations of computer representations • Start, Tail, Head, Next • Vert, Tail, Head, Next

20. Chapter 5 – Functions5.1 Functions5.2 Functions for CS • Definition • Compositions • Special functions • Everywhere defined • Onto • 1 – 1 • Invertible functions • Cryptology – Substitution code • Special Functions for Computer Science

21. Chapter 5 – Functions5.2 Functions for CS • Special Functions for Computer Science • Fuzzy sets • Degree to which an element is in a set • Fuzzy set operations • Degree of membership of an element in a set

22. Chapter 5 – Functions5.3 Growth of Functions • Show f is O(g) • Show f and g have the same order • Theta-classes

23. Chapter 5 – Functions5.4 Permutations • Definition • Compositions, Inverses • Cycles • Transpositions (even, odd permutations) • Ex: Writeas a product of transpositions • Cryptology – transposition codes and keyword columnar transpositions

24. Ch. 6 – Order Rel & Structures6.1 Partially ordered sets • Reflexive, Antisymmetric, Transitive • Hassediagrams • Topological sortings • Isomorphism

25. Ch. 6 – Order Rel & Structures 6.2 Extremal Elements • Maximal • Minimal • Greatest • Least • Upper Bounds (LUB) • Lower Bounds (GLB)

26. Ch. 6 – Order Rel & Structures 6.3 Lattices6.4 Boolean Algebras • Lattice – POSET where every 2 element subset has LUB and GLB • Boolean Algebra – Lattice that is isomorphic to Bn for some n in Z+

27. Ch. 6 – Order Rel & Structures 6.5 Functions on Boolean Algebras • Truth tables of functions • Schematics

28. Chapter 7 – Trees7.1 Trees7.2 Labeled Trees • Terminology • Constructing Trees • Computer Representations

29. Chapter 7 – Trees7.3 Tree Searching • Algorithms • Preorder (and Polish notation) • Postorder (and Reverse Polish notation) • Inorder (and infix notation) • Finding the binary representation of a tree • Searching non-binary trees

30. Chapter 7 – Trees7.4 Undirected Trees7.5 Minimal Spanning Trees • Spanning tree (Prim – 7.4) • Minimal spanning tree (Prim, Kruskal – 7.5)

31. Chapter 8 – Graphs8.1 Topics in graph theory • Definition (Set of vertices, edges, and function) • Terminology • Special Graphs • Un, Kn, Ln, Regular Graphs • Subgraphs (delete edges) • Quotient Graphs (merge equivalence classes)

32. Chapter 8 – Graphs8.2 Euler Paths and Circuits8.3 Hamiltonian Paths and Circuits • Euler – edges • Fleury’s Algorithm • Hamilton – vertices • Existence Theorems

33. Chapter 10 – Finite State Machines10.1 Languages • Phrase Structure Grammars (V, S, v0, relation) • Determining if an element is in the language • Describing a language • Derivation trees • Types (0 – 3)

34. Chapter 10 – Finite State Machines10.2 Presentations • BNF Form • Syntax Diagrams

35. Chapter 10 – Finite State Machines10.3 Finite State Machines • Terminology • States • State Transitions • Tasks • Describe functions given state transition table • Describe state transition table given functions • RM and digraphs