1. MATH 370 Final Review Chapter 5-10

2. 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

3. …topics to know well Ch. 9: Graphs Special graphs: K n, C n, W n, Q n, K m,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

4. 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)  R 4, find path length 4 in R Maximal, minimal, greatest, least, glb, lub Hasse Compatible total order

5. …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

6. Will be provided, so you can use them to calculate/ prove other things Ch. 5 Binomial formula Ch. 7 7.1: ∑ ar i = … 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

7. …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 Cor 3: If G conn, planar simple, with no circuits length 3, then e≤2v-4 Thm. 2: A graph is nonplanar iff it contains a subgraph homeomorphic 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…

8. Skip these Ch. 6: Derangements 8.2: databases Any proofs in 9.6 Proofs in ch. 10