CSCI 115 Course Review

Chapter 1 – Fundamentals 1.1 Sets and Subsets Set equality Special sets (Z, Z+, Q, R, {}) Power sets Cardinality Subset notation and meaning

Chapter 1 – Fundamentals 1.2 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|

Chapter 1 – Fundamentals 1.3 Sequences Definition Characteristic Function (and computer representations) Countable and Uncountable Sets Regular Expressions

Chapter 1 – Fundamentals 1.4 Division in the Integers Prime numbers Divides (a | b) GCD LCM Number bases Cryptology – Sir Francis Bacon’s code

Chapter 1 – Fundamentals 1.5 Matrices Terminology Operations (add, sub, multiply) Boolean Matrices and Operations Join (or) Meet (and) Boolean Product

Chapter 1 – Fundamentals 1.6 Mathematical Structures Objects Operations Possible existence of identity Other properties (Associative, commutative, etc.)

Chapter 2 – Logic 2.1 Propositions and Log Ops Statements Logical operators (and, or, not) Truth Tables Quantifiers Universal Existential

Chapter 2 – Logic 2.2 Conditional Statements Biconditional Converse Inverse Contrapositive Standard Truth Tables

Chapter 2 – Logic 2.3 Methods of Proof 2.4 Mathematical Induction Direct Proof Contradiction Other tips / techniques (even / odd, etc.) Mathematical Induction

Chapter 3 – Counting 3.1 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

Chapter 3 – Counting 3.4 Elements of Probability Sample Spaces and Events Probability spaces Equally likely outcomes Expected values

Chapter 3 – Counting 3.5 Recurrence Relations Techniques ‘Eyeball’ Backtracking Linear Homogeniety

Chapter 4 – Relations and Digraphs 4.1 Product Sets and Partitions Ex. R x R Partitions

Chapter 4 – Relations and Digraphs 4.2 Relations and Digraphs Relations – What are they? Domains Ranges Relation Element Subset Representations Ordered Pairs Matrix Digraph Restriction to a subset

Chapter 4 – Relations and Digraphs 4.3 Paths in Relations and Digraphs Compositions Relations

Chapter 4 – Relations and Digraphs 4.4 Properties of Relations Reflexive Irreflexive Symmetric Asymmetric Antisymmetric Transitive

Chapter 4 – Relations and Digraphs 4.5 Equivalence Relations Equivalence Relation: Ref, Symm, Trans Ex: R on Z+ by aRb iff a=b(mod 2) Equivalence Classes A/R (Partition)

Chapter 4 – Relations and Digraphs 4.6 Computer Representations Linked Lists Different implementations of computer representations Start, Tail, Head, Next Vert, Tail, Head, Next

Chapter 5 – Functions 5.1 Functions 5.2 Functions for CS Definition Compositions Special functions Everywhere defined Onto 1 – 1 Invertible functions Cryptology – Substitution code Special Functions for Computer Science

Chapter 5 – Functions 5.2 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

Chapter 5 – Functions 5.3 Growth of Functions Show f is O(g) Show f and g have the same order Theta-classes

Chapter 5 – Functions 5.4 Permutations Definition Compositions, Inverses Cycles Transpositions (even, odd permutations) Ex: Write as a product of transpositions Cryptology – transposition codes and keyword columnar transpositions

Ch. 6 – Order Rel & Structures 6.1 Partially ordered sets Reflexive, Antisymmetric, Transitive Hasse diagrams Topological sortings Isomorphism

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

Ch. 6 – Order Rel & Structures 6.3 Lattices 6.4 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+

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

Chapter 7 – Trees 7.1 Trees 7.2 Labeled Trees Terminology Constructing Trees Computer Representations

Chapter 7 – Trees 7.3 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

Chapter 7 – Trees 7.4 Undirected Trees 7.5 Minimal Spanning Trees Spanning tree (Prim – 7.4) Minimal spanning tree (Prim, Kruskal – 7.5)

Chapter 8 – Graphs 8.1 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)

Chapter 8 – Graphs 8. 2 Euler Paths and Circuits 8 Chapter 8 – Graphs 8.2 Euler Paths and Circuits 8.3 Hamiltonian Paths and Circuits Euler – edges Fleury’s Algorithm Hamilton – vertices Existence Theorems

Chapter 10 – Finite State Machines 10.1 Languages Phrase Structure Grammars (V, S, v0, relation) Determining if an element is in the language Describing a language Derivation trees Types (0 – 3)

Chapter 10 – Finite State Machines 10.2 Presentations BNF Form Syntax Diagrams

Chapter 10 – Finite State Machines 10.3 Finite State Machines Terminology States State Transitions Tasks Describe functions given state transition table Describe state transition table given functions RM and digraphs