Midterm 2 cheat sheet, Cheat Sheet of Mathematics

Download Midterm 2 cheat sheet and more Cheat Sheet Mathematics in PDF only on Docsity! MIDTERM 2 REFERENCE SHEET (MATH 61, SPRING 2023) DO NOT TURN IN Below let G be a simple graph and n, k ∈ Z≥0, unless otherwise specified: • (Binomial Theorem) (a+ b)n = n∑ k=0 ( n k ) an−kbk – This implies that 2n = n∑ k=0 ( n k ) . • P (n, k) = n! (n−k)! for k ≤ n – ways to select k-permutations from an n element set • C(n, k) = ( n k ) = n! (n−k)!k! for k ≤ n – ways to select k-combinations from an n element set • ( n+k−1 n ) – ways to make n unordered selections from a set containing k elements, allowing repetition • ( n+1 k ) = ( n k ) + ( n k−1 ) – different ways to choose k-combinations of an n+ 1-element set • n∑ i=k ( i k ) = ( n+ 1 k + 1 ) • (Generalized Pigeonhole Principle) For X,Y finite, let f : X → Y where ⌈ |X| |Y | ⌉ = k. Then there is a k-subset {x1, x2, . . . , xk} ⊆ X such that f(x1) = . . . = f(xk). • (Linear Homogeneous Recurrence Relation) c0an = c1an−1 + c2an−2 with Constant Coefficients (LHRC), where c0, c1, c2 ∈ R and n ≥ n0 for some n0 ∈ Z. – when characteristic polynomial c0t 2 − c1t − c2 has roots r1 ̸= r2: an = brn1 + drn2 for some b, d ∈ R – when characteristic polynomial c0t 2−c1t−c2 has repeated root r: an = brn+dnrn for some b, d ∈ R • (Handshake Lemma) ∑ v∈V (G) δ(v) = 2 · |E(G)| – This implies there are always an even number of odd degree vertices in G. • G has an Euler cycle if and only if G is connected and each vertex of G has even degree • (Euler’s Theorem) If G is a connected and planar with f faces, e edges, v vertices, then f = e− v + 2. • G is bipartite if and only if G contains no cycles of odd length. • (Kuratowski’s theorem) G is non-planar if and only if G contains a subgraph home- omorphic to K3,3 or K5 Note: you do not need to re-prove statements, principles, and theorems given in class. You need only justify why they apply to the situations where you are using them (unless explicitly stated otherwise). 1