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CSCI 115 Course Review
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Chapter 1 – Fundamentals 1.1 Sets and Subsets
Set equality Special sets (Z, Z+, Q, R, {}) Power sets Cardinality Subset notation and meaning
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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|
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Chapter 1 – Fundamentals 1.3 Sequences
Definition Characteristic Function (and computer representations) Countable and Uncountable Sets Regular Expressions
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Chapter 1 – Fundamentals 1.4 Division in the Integers
Prime numbers Divides (a | b) GCD LCM Number bases Cryptology – Sir Francis Bacon’s code
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Chapter 1 – Fundamentals 1.5 Matrices
Terminology Operations (add, sub, multiply) Boolean Matrices and Operations Join (or) Meet (and) Boolean Product
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Chapter 1 – Fundamentals 1.6 Mathematical Structures
Objects Operations Possible existence of identity Other properties (Associative, commutative, etc.)
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Chapter 2 – Logic 2.1 Propositions and Log Ops
Statements Logical operators (and, or, not) Truth Tables Quantifiers Universal Existential
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Chapter 2 – Logic 2.2 Conditional Statements
Biconditional Converse Inverse Contrapositive Standard Truth Tables
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Chapter 2 – Logic 2.3 Methods of Proof 2.4 Mathematical Induction
Direct Proof Contradiction Other tips / techniques (even / odd, etc.) Mathematical Induction
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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
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Chapter 3 – Counting 3.4 Elements of Probability
Sample Spaces and Events Probability spaces Equally likely outcomes Expected values
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Chapter 3 – Counting 3.5 Recurrence Relations
Techniques ‘Eyeball’ Backtracking Linear Homogeniety
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Chapter 4 – Relations and Digraphs 4.1 Product Sets and Partitions
Ex. R x R Partitions
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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
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Chapter 4 – Relations and Digraphs 4.3 Paths in Relations and Digraphs
Compositions Relations
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Chapter 4 – Relations and Digraphs 4.4 Properties of Relations
Reflexive Irreflexive Symmetric Asymmetric Antisymmetric Transitive
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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)
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Chapter 4 – Relations and Digraphs 4.6 Computer Representations
Linked Lists Different implementations of computer representations Start, Tail, Head, Next Vert, Tail, Head, Next
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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
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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
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Chapter 5 – Functions 5.3 Growth of Functions
Show f is O(g) Show f and g have the same order Theta-classes
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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
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Ch. 6 – Order Rel & Structures 6.1 Partially ordered sets
Reflexive, Antisymmetric, Transitive Hasse diagrams Topological sortings Isomorphism
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Ch. 6 – Order Rel & Structures 6.2 Extremal Elements
Maximal Minimal Greatest Least Upper Bounds (LUB) Lower Bounds (GLB)
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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+
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Ch. 6 – Order Rel & Structures 6.5 Functions on Boolean Algebras
Truth tables of functions Schematics
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Chapter 7 – Trees 7.1 Trees 7.2 Labeled Trees
Terminology Constructing Trees Computer Representations
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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
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Chapter 7 – Trees 7.4 Undirected Trees 7.5 Minimal Spanning Trees
Spanning tree (Prim – 7.4) Minimal spanning tree (Prim, Kruskal – 7.5)
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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)
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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
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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)
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Chapter 10 – Finite State Machines 10.2 Presentations
BNF Form Syntax Diagrams
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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
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