 Chapter 1: Joy
 Chapter 10: Sets I: Introduction, Subsets
 Chapter 11: Sets I: Introduction, Subsets
 Chapter 12: Sets II: Operations
 Chapter 13: Combinatorial Proof: Two Examples
 Chapter 14: Relations
 Chapter 15: Equivalence Relations
 Chapter 16: Partitions
 Chapter 17: Binomial Coefficients
 Chapter 18: Counting Multisets
 Chapter 19: InclusionExclusion
 Chapter 2: Speaking (and Writing) of Mathematics
 Chapter 20: Contradiction
 Chapter 21: Smallest Counterexample
 Chapter 22: Induction
 Chapter 23: Recurrence Relations
 Chapter 24: Functions
 Chapter 25: The Pigeonhole Principle
 Chapter 26: Composition
 Chapter 27: Permutations
 Chapter 28: Symmetry
 Chapter 29: Assorted Notation
 Chapter 3: Definition
 Chapter 30: Sample Space
 Chapter 31: Events
 Chapter 32: Conditional Probability and Independence
 Chapter 33: Random Variables
 Chapter 34: Expectation
 Chapter 35: Dividing
 Chapter 36: Greatest Common Divisor
 Chapter 37: Modular Arithmetic
 Chapter 38: The Chinese Remainder Theorem
 Chapter 39: Factoring
 Chapter 4: Theorem
 Chapter 40: Groups
 Chapter 41: Group Isomorphism The Same?
 Chapter 42: Subgroups
 Chapter 43: Fermats Little Theorem
 Chapter 44: Public Key Cryptography I: Introduction The Problem: Private Communication in Public
 Chapter 45: Public Key Cryptography II: Rabins Method
 Chapter 46: Public Key Cryptography III: RSA
 Chapter 47: Fundamentals of Graph Theory
 Chapter 48: Subgraphs
 Chapter 49: Connection
 Chapter 5: Proof
 Chapter 50: Trees
 Chapter 51: Eulerian Graphs
 Chapter 52: Coloring
 Chapter 53: Planar Graphs
 Chapter 54: Fundamentals of Partially Ordered Sets
 Chapter 55: Max and Min
 Chapter 56: Linear Orders
 Chapter 57: Linear Extensions
 Chapter 58: Dimension
 Chapter 59: Lattices
 Chapter 6: Counterexample
 Chapter 7: Boolean Algebra
 Chapter 8: Lists
 Chapter 9: Factorial
 Chapter Chapter 1: Fundamentals
 Chapter Chapter 10: Partially Ordered Sets
 Chapter Chapter 2: Collections
 Chapter Chapter 3: Counting and Relations
 Chapter Chapter 4: More Proof
 Chapter Chapter 5: Functions
 Chapter Chapter 6: Probability
 Chapter Chapter 7: Number Theory
 Chapter Chapter 8: Algebra
 Chapter Chapter 9: Graphs
Mathematics: A Discrete Introduction 3rd Edition  Solutions by Chapter
Full solutions for Mathematics: A Discrete Introduction  3rd Edition
ISBN: 9780840049421
Mathematics: A Discrete Introduction  3rd Edition  Solutions by Chapter
Get Full SolutionsSince problems from 69 chapters in Mathematics: A Discrete Introduction have been answered, more than 3135 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 69. The full stepbystep solution to problem in Mathematics: A Discrete Introduction were answered by , our top Math solution expert on 03/15/18, 06:06PM. This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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