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CSCI 115
Course Review
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Chapter 1 – Fundamentals1.1 Sets and Subsets
• Set equality• Special sets (Z, Z+, Q, R, {})• Power sets• Cardinality• Subset notation and meaning
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Chapter 1 – 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|
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Chapter 1 – Fundamentals1.3 Sequences
• Definition• Characteristic Function (and computer
representations)• Countable and Uncountable Sets• Regular Expressions
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Chapter 1 – Fundamentals1.4 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 – Fundamentals1.5 Matrices
• Terminology• Operations (add, sub, multiply)• Boolean Matrices and Operations
– Join (or)– Meet (and)– Boolean Product
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Chapter 1 – Fundamentals1.6 Mathematical Structures
• Structure– Objects– Operations– Possible existence of identity– Other properties (Associative, commutative,
etc.)
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Chapter 2 – Logic2.1 Propositions and Log Ops
• Statements• Logical operators (and, or, not)• Truth Tables• Quantifiers
– Universal– Existential
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Chapter 2 – Logic2.2 Conditional Statements
• Conditional• Biconditional• Converse• Inverse• Contrapositive• Standard Truth Tables
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Chapter 2 – Logic2.3 Methods of Proof
2.4 Mathematical Induction
• Direct Proof• Contradiction• Other tips / techniques
– (even / odd, etc.)
• Mathematical Induction
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Chapter 3 – Counting3.1 Permutations and 3.2 Combinations
• Principle of Counting
• Permutations:
• Combinations:
!
( )!
n
n r
!
!( )!
n
r n r
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Chapter 3 – Counting3.4 Elements of Probability
• Sample Spaces and Events• Probability spaces• Equally likely outcomes• Expected values
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Chapter 3 – Counting3.5 Recurrence Relations
• Techniques– ‘Eyeball’– Backtracking– Linear Homogeneity
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Chapter 4 – Relations and Digraphs4.1 Product Sets and Partitions
• Product Sets• Partitions
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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
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Chapter 4 – Relations and Digraphs 4.3 Paths in Relations and Digraphs
• Paths– Compositions– Relations
• *, , nR R R
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Chapter 4 – Relations and Digraphs4.4 Properties of Relations
• Reflexive• Irreflexive• Symmetric• Asymmetric• Antisymmetric• Transitive
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Chapter 4 – Relations and Digraphs4.5 Equivalence Relations
• Equivalence Relation: Ref, Symm, Trans• Equivalence Classes• A/R (Partition)
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Chapter 4 – Relations and Digraphs4.6 Computer Representations
• Linked Lists• Different implementations of computer
representations• Start, Tail, Head, Next• Vert, Tail, Head, Next
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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
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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
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Chapter 5 – Functions5.3 Growth of Functions
• Show f is O(g)• Show f and g have the same order• Theta-classes
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Chapter 5 – Functions5.4 Permutations
• Definition• Compositions, Inverses• Cycles• Transpositions (even, odd permutations)• Cryptology – transposition codes and
keyword columnar transpositions
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Ch. 6 – Order Rel & Structures6.1 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 – Trees7.1 Trees
7.2 Labeled Trees
• Terminology• Constructing Trees• Computer Representations
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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
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Chapter 7 – Trees7.4 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 – 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)
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Chapter 8 – Graphs8.2 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 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)
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Chapter 10 – Finite State Machines10.2 Presentations
• BNF Form• Syntax Diagrams
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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