MATH 370 Final Review

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# MATH 370 Final Review - PowerPoint PPT Presentation

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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### 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
• 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

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

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 apply

Ch. 9:

• Thm. 2: undirected graph has an even # of odd degree
• Calculate deg, deg-, deg+
• 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

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

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
• Ch. 6: Derangements
• 8.2: databases
• Any proofs in 9.6
• Proofs in ch. 10