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MATH 370 Final Review. Chapter 5-10. Know this well. Chapter 5 and 6: Counting/ Probability Basic counting P( n,r ), C( n,r ), C(n-1+r,r),… Basic probability, expected value Ch . 7: Recurrence Finding solutions Proving these are solutions Ch . 8: Relations Relation, function defs

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know this well
Know this well

Chapter 5 and 6: Counting/ Probability

  • Basic counting
  • P(n,r), C(n,r), C(n-1+r,r),…
  • Basic probability, expected value

Ch. 7: Recurrence

  • Finding solutions
  • Proving these are solutions

Ch. 8: Relations

  • Relation, function defs
  • Def of R,S, A, T
  • Def of divides
  • Def of a=b mod m…
  • Def of Equiv Relation: RST
  • Def of PO: RAT
  • Def of comparable
  • Def of total order
topics to know well
…topics to know well

Ch. 9: Graphs

  • Special graphs: Kn, Cn, Wn, Qn, Km,n
  • Prove:
  • Bipartitite or not
  • Isomorphic or not
  • Planar or not
  • 9.8: Thm 1- chromatic # of planar graph ≤4

Ch. 10: Trees

  • Def of tree, rooted tree

Basic Proof Methods

  • Direct proofs, utilizing definitions (ex: show R is transitive using the definition– Assume aRb and bRc. Show aRc.)
  • Indirect (contrapositive) and By Contradiction
  • Cases
  • Induction
  • Disproving, by using a counterexample
techniques to apply
Techniques to apply

You won’t need to state these definitions, but be able to do:

Ch. 7:

  • 7.5: |AUAUA|=…

Ch. 8:

  • Matrices and digraphs and graphs
  • Do relations have certain properties: RSAT
  • Find closures
  • For (a,b) R4, find path length 4 in R
  • Maximal, minimal, greatest, least, glb, lub
  • Hasse
  • Compatible total order
techniques to apply1
…Techniques to apply

Ch. 9:

  • Thm. 2: undirected graph has an even # of odd degree
  • Calculate deg, deg-, deg+
  • Adjacency tables and matrices
  • Paths
  • Strong and weakly connected
  • Counting paths of length l
  • Euler and Hamilton paths and circuits
  • Conditions for Euler paths and circuits (not for Hamilton)
  • Chromatic number of special graphs

Ch. 10:

  • Determine if a tree or not
  • 10.3: pre, in, and postorder and Infix, prefix, postfix notation
  • 10.4: find spanning tree
  • 10.5: find minimum spanning tree
will be provided so you can use them to calculate prove other things
Will be provided, so you can use them to calculate/ prove other things

Ch. 5

  • Binomial formula

Ch. 7

  • 7.1: ∑ari= …
  • 7.2: Thm 1 and 2 on how to find solutions to recurrence relations
  • N(P1’P2’…) formula
  • SοR definition; R n+1 = R n ο R; M S ο R = M R ο M S

Ch. 8

  • These statements will be given, so you may need to prove them:
    • 8.1: Thm. 1. R is transitive implies R n R
    • Thm. 2: 8.4: R* = U R n is the transitive closure of R
    • R* is transitive
will be provided so you can use them to calculate prove other things1
…Will be provided, so you can use them to calculate/ prove other things

Ch. 9:

  • 9.2: Thm. 1 Handshaking: 2e= sum of deg(v)…
  • Euler: r=e-v+2
  • Cor 1: If G connected, planar, simple, e≤ 3v-6
  • Cor3: If G conn, planar simple, with no circuits length 3, then e≤2v-4
  • Thm. 2: A graph is nonplanariff it contains a subgraphhomeomorphic to K3,3 or K5.

Ch. 10:

  • 10.1: Thm 2: tree with n vertices has n-1 edges
  • Thm 3: full m-ary tree with I internal vertices contains n=mi+1 vertices
  • 10.1: Thm 4 (p.691): A full m-ary tree with n vertices has i=(n-1)/m internal vertices…
skip these
Skip these
  • Ch. 6: Derangements
  • 8.2: databases
  • Any proofs in 9.6
  • Proofs in ch. 10