# csci 115 course review. chapter 1 – fundamentals 1.1 sets and subsets set equality special sets...

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CIS 130

CSCI 115Course ReviewChapter 1 Fundamentals1.1 Sets and SubsetsSet equalitySpecial sets (Z, Z+, Q, R, {})Power setsCardinalitySubset notation and meaningChapter 1 Fundamentals1.2 Operations on SetsUnionIntersectionComplementSymmetric DifferenceAddition PrinciplesFor 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 SequencesDefinitionCharacteristic Function (and computer representations)Countable and Uncountable SetsRegular ExpressionsChapter 1 Fundamentals1.4 Division in the IntegersPrime numbersDivides (a | b)GCDLCMNumber basesCryptology Sir Francis Bacons codeChapter 1 Fundamentals1.5 MatricesTerminologyOperations (add, sub, multiply)Boolean Matrices and OperationsJoin (or)Meet (and)Boolean ProductChapter 1 Fundamentals1.6 Mathematical StructuresStructureObjectsOperationsPossible existence of identityOther properties (Associative, commutative, etc.)Chapter 2 Logic2.1 Propositions and Log OpsStatementsLogical operators (and, or, not)Truth TablesQuantifiersUniversalExistentialChapter 2 Logic2.2 Conditional StatementsConditionalBiconditionalConverseInverseContrapositiveStandard Truth TablesChapter 2 Logic2.3 Methods of Proof2.4 Mathematical InductionDirect ProofContradictionOther tips / techniques(even / odd, etc.)Mathematical InductionChapter 3 Counting3.1 Permutations and 3.2 CombinationsPrinciple of Counting

Permutations:

Combinations:

Chapter 3 Counting3.4 Elements of ProbabilitySample Spaces and EventsProbability spacesEqually likely outcomesExpected valuesChapter 3 Counting3.5 Recurrence RelationsTechniquesEyeballBacktrackingLinear HomogeneityChapter 4 Relations and Digraphs4.1 Product Sets and PartitionsProduct SetsPartitionsChapter 4 Relations and Digraphs4.2 Relations and DigraphsRelations What are they?DomainsRangesRelationElementSubsetRepresentationsOrdered PairsMatrixDigraphRestriction to a subsetChapter 4 Relations and Digraphs 4.3 Paths in Relations and DigraphsPathsCompositionsRelations

Chapter 4 Relations and Digraphs4.4 Properties of RelationsReflexiveIrreflexiveSymmetricAsymmetricAntisymmetricTransitiveChapter 4 Relations and Digraphs4.5 Equivalence RelationsEquivalence Relation: Ref, Symm, TransEquivalence ClassesA/R (Partition)Chapter 4 Relations and Digraphs4.6 Computer RepresentationsLinked ListsDifferent implementations of computer representationsStart, Tail, Head, NextVert, Tail, Head, NextChapter 5 Functions5.1 Functions5.2 Functions for CSDefinitionCompositionsSpecial functionsEverywhere definedOnto1 1Invertible functionsCryptology Substitution codeSpecial Functions for Computer ScienceChapter 5 Functions5.2 Functions for CSSpecial Functions for Computer ScienceFuzzy setsDegree to which an element is in a setFuzzy set operationsDegree of membership of an element in a setChapter 5 Functions5.3 Growth of FunctionsShow f is O(g)Show f and g have the same orderTheta-classesChapter 5 Functions5.4 PermutationsDefinitionCompositions, InversesCyclesTranspositions (even, odd permutations)Cryptology transposition codes and keyword columnar transpositionsCh. 6 Order Rel & Structures6.1 Partially ordered setsReflexive, Antisymmetric, TransitiveHasse diagramsTopological sortingsIsomorphismCh. 6 Order Rel & Structures 6.2 Extremal ElementsMaximalMinimalGreatestLeastUpper Bounds (LUB)Lower Bounds (GLB)Ch. 6 Order Rel & Structures 6.3 Lattices6.4 Boolean AlgebrasLattice 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 AlgebrasTruth tables of functionsSchematicsChapter 7 Trees7.1 Trees7.2 Labeled TreesTerminologyConstructing TreesComputer RepresentationsChapter 7 Trees7.3 Tree SearchingAlgorithmsPreorder (and Polish notation)Postorder (and Reverse Polish notation)Inorder (and infix notation)Finding the binary representation of a treeSearching non-binary treesChapter 7 Trees7.4 Undirected Trees7.5 Minimal Spanning TreesSpanning tree (Prim 7.4)

Minimal spanning tree (Prim, Kruskal 7.5)Chapter 8 Graphs8.1 Topics in graph theoryDefinition (Set of vertices, edges, and function)TerminologySpecial GraphsUn, Kn, Ln, Regular GraphsSubgraphs (delete edges)Quotient Graphs (merge equivalence classes)Chapter 8 Graphs8.2 Euler Paths and Circuits8.3 Hamiltonian Paths and CircuitsEuler edgesFleurys AlgorithmHamilton verticesExistence TheoremsChapter 10 Finite State Machines10.1 LanguagesPhrase Structure Grammars (V, S, v0, relation)Determining if an element is in the languageDescribing a languageDerivation treesTypes (0 3)Chapter 10 Finite State Machines10.2 PresentationsBNF FormSyntax DiagramsChapter 10 Finite State Machines10.3 Finite State MachinesTerminologyStatesState Transitions

TasksDescribe functions given state transition tableDescribe state transition table given functionsRM and digraphs