CSCI 115

Course Review

Chapter 1 – Fundamentals1.1 Sets and Subsets

• Set equality• Special sets (Z, Z+, Q, R, {})• Power sets• Cardinality• Subset notation and meaning

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|

Chapter 1 – Fundamentals1.3 Sequences

• Definition• Characteristic Function (and computer

representations)• Countable and Uncountable Sets• Regular Expressions

Chapter 1 – Fundamentals1.4 Division in the Integers

• Prime numbers• Divides (a | b)• GCD• LCM• Number bases• Cryptology – Sir Francis Bacon’s code

Chapter 1 – Fundamentals1.5 Matrices

• Terminology• Operations (add, sub, multiply)• Boolean Matrices and Operations

– Join (or)– Meet (and)– Boolean Product

Chapter 1 – Fundamentals1.6 Mathematical Structures

• Structure– Objects– Operations– Possible existence of identity– Other properties (Associative, commutative,

etc.)

Chapter 2 – Logic2.1 Propositions and Log Ops

• Statements• Logical operators (and, or, not)• Truth Tables• Quantifiers

– Universal– Existential

Chapter 2 – Logic2.2 Conditional Statements

• Conditional• Biconditional• Converse• Inverse• Contrapositive• Standard Truth Tables

Chapter 2 – Logic2.3 Methods of Proof

2.4 Mathematical Induction

• Direct Proof• Contradiction• Other tips / techniques

– (even / odd, etc.)

• Mathematical Induction

Chapter 3 – Counting3.1 Permutations and 3.2 Combinations

• Principle of Counting

• Permutations:

• Combinations:

!

( )!

n

n r

!

!( )!

n

r n r

Chapter 3 – Counting3.4 Elements of Probability

• Sample Spaces and Events• Probability spaces• Equally likely outcomes• Expected values

Chapter 3 – Counting3.5 Recurrence Relations

• Techniques– ‘Eyeball’– Backtracking– Linear Homogeneity

Chapter 4 – Relations and Digraphs4.1 Product Sets and Partitions

• Product Sets• Partitions

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

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

• Paths– Compositions– Relations

• *, , nR R R

Chapter 4 – Relations and Digraphs4.4 Properties of Relations

• Reflexive• Irreflexive• Symmetric• Asymmetric• Antisymmetric• Transitive

Chapter 4 – Relations and Digraphs4.5 Equivalence Relations

• Equivalence Relation: Ref, Symm, Trans• Equivalence Classes• A/R (Partition)

Chapter 4 – Relations and Digraphs4.6 Computer Representations

• Linked Lists• Different implementations of computer

representations• Start, Tail, Head, Next• Vert, Tail, Head, Next

Chapter 5 – Functions5.1 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 – 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

Chapter 5 – Functions5.3 Growth of Functions

• Show f is O(g)• Show f and g have the same order• Theta-classes

Chapter 5 – Functions5.4 Permutations

• Definition• Compositions, Inverses• Cycles• Transpositions (even, odd permutations)• Cryptology – transposition codes and

keyword columnar transpositions

Ch. 6 – Order Rel & Structures6.1 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 – Trees7.1 Trees

7.2 Labeled Trees

• Terminology• Constructing Trees• Computer Representations

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

Chapter 7 – Trees7.4 Undirected Trees

7.5 Minimal Spanning Trees

• Spanning tree (Prim – 7.4)

• Minimal spanning tree (Prim, Kruskal – 7.5)

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)

Chapter 8 – Graphs8.2 Euler Paths and Circuits

8.3 Hamiltonian Paths and Circuits

• Euler – edges• Fleury’s Algorithm• Hamilton – vertices• Existence Theorems

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)

Chapter 10 – Finite State Machines10.2 Presentations

• BNF Form• Syntax Diagrams

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