 Chapter 1.1:
 Chapter 1.1: Variables
 Chapter 1.2:
 Chapter 1.2: The Language of Sets
 Chapter 1.3:
 Chapter 1.3: The Language of Relations and Functions
 Chapter 10.1:
 Chapter 10.1: Graphs: Definitions and Basic Properties
 Chapter 10.2:
 Chapter 10.2: Trails, Paths, and Circuits
 Chapter 10.3:
 Chapter 10.3: Matrix Representations of Graphs
 Chapter 10.4:
 Chapter 10.4: Isomorphisms of Graphs
 Chapter 10.5:
 Chapter 10.5: Trees
 Chapter 10.6:
 Chapter 10.6: Rooted Trees
 Chapter 10.7:
 Chapter 10.7: Spanning Trees and Shortest Paths
 Chapter 11.1:
 Chapter 11.1: RealValued Functions of a Real Variable and Their Graphs
 Chapter 11.2:
 Chapter 11.2: O, , and Notations
 Chapter 11.3:
 Chapter 11.3: Application: Analysis of Algorithm Efficiency I
 Chapter 11.4:
 Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders
 Chapter 11.5:
 Chapter 11.5: Application: Analysis of Algorithm Efficiency II
 Chapter 12.1:
 Chapter 12.1: Formal Languages and Regular Expressions
 Chapter 12.2:
 Chapter 12.2: FiniteState Automata
 Chapter 12.3:
 Chapter 12.3: Simplifying FiniteState Automata
 Chapter 2.1:
 Chapter 2.1: Logical Form and Logical Equivalence
 Chapter 2.2:
 Chapter 2.2: Conditional Statements
 Chapter 2.3:
 Chapter 2.3: Valid and Invalid Arguments
 Chapter 2.4:
 Chapter 2.4: Application: Digital Logic Circuits
 Chapter 2.5:
 Chapter 2.5: Application: Number Systems and Circuits for Addition
 Chapter 3.1: Predicates and Quantified Statements I
 Chapter 3.2:
 Chapter 3.2: Predicates and Quantified Statements II
 Chapter 3.3:
 Chapter 3.3: Statements with Multiple Quantifiers
 Chapter 3.4:
 Chapter 3.4: Arguments with Quantified Statements
 Chapter 4.1:
 Chapter 4.1: Direct Proof and Counterexample I: Introduction
 Chapter 4.2:
 Chapter 4.2: Direct Proof and Counterexample II: Rational Numbers
 Chapter 4.3:
 Chapter 4.3: Direct Proof and Counterexample III: Divisibility
 Chapter 4.4:
 Chapter 4.4: Direct Proof and Counterexample IV: Division into Cases and the QuotientRemainder Theorem
 Chapter 4.5:
 Chapter 4.5: Direct Proof and Counterexample V: Floor and Ceiling
 Chapter 4.6:
 Chapter 4.6: Indirect Argument: Contradiction and Contraposition
 Chapter 4.7:
 Chapter 4.7: Indirect Argument: Two Classical Theorems
 Chapter 4.8:
 Chapter 4.8: Application: Algorithms
 Chapter 5.1:
 Chapter 5.1: Sequences
 Chapter 5.2:
 Chapter 5.2: Mathematical Induction I
 Chapter 5.3:
 Chapter 5.3: Mathematical Induction II
 Chapter 5.4:
 Chapter 5.4: Strong Mathematical Induction and the WellOrdering Principle for the Integers
 Chapter 5.5:
 Chapter 5.5: Application: Correctness of Algorithms
 Chapter 5.6:
 Chapter 5.6: Defining Sequences Recursively
 Chapter 5.7:
 Chapter 5.7: Solving Recurrence Relations by Iteration
 Chapter 5.8:
 Chapter 5.8: SecondOrder Linear Homogeneous Recurrence Relations with Constant Coefficients
 Chapter 5.9:
 Chapter 5.9: General Recursive Definitions and Structural Induction
 Chapter 6.1:
 Chapter 6.1: Set Theory: Definitions and the Element Method of Proof
 Chapter 6.2:
 Chapter 6.2: Properties of Sets
 Chapter 6.3:
 Chapter 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras
 Chapter 6.4:
 Chapter 6.4: Boolean Algebras, Russells Paradox, and the Halting Problem
 Chapter 7.1:
 Chapter 7.1: Functions Defined on General Sets
 Chapter 7.2:
 Chapter 7.2: OnetoOne and Onto, Inverse Functions
 Chapter 7.3:
 Chapter 7.3: Composition of Functions
 Chapter 7.4:
 Chapter 7.4: Cardinality with Applications to Computability
 Chapter 8.1:
 Chapter 8.1: Relations on Sets
 Chapter 8.2:
 Chapter 8.2: Reflexivity, Symmetry, and Transitivity
 Chapter 8.3:
 Chapter 8.3: Equivalence Relations
 Chapter 8.4:
 Chapter 8.4: Modular Arithmetic with Applications to Cryptography
 Chapter 8.5:
 Chapter 8.5: Partial Order Relations
 Chapter 9.1:
 Chapter 9.1: Introduction
 Chapter 9.2:
 Chapter 9.2: Possibility Trees and the Multiplication Rule
 Chapter 9.3:
 Chapter 9.3: Counting Elements of Disjoint Sets: The Addition Rule
 Chapter 9.4:
 Chapter 9.4: The Pigeonhole Principle
 Chapter 9.5:
 Chapter 9.5: Counting Subsets of a Set: Combinations
 Chapter 9.6:
 Chapter 9.6: rCombinations with Repetition Allowed
 Chapter 9.7:
 Chapter 9.7: Pascals Formula and the Binomial Theorem
 Chapter 9.8:
 Chapter 9.8: Probability Axioms and Expected Value
 Chapter 9.9:
 Chapter 9.9: Conditional Probability, Bayes Formula, and Independent Events
Discrete Mathematics with Applications 4th Edition  Solutions by Chapter
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Discrete Mathematics with Applications  4th Edition  Solutions by Chapter
Get Full SolutionsThis expansive textbook survival guide covers the following chapters: 131. Since problems from 131 chapters in Discrete Mathematics with Applications have been answered, more than 90673 students have viewed full stepbystep answer. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. The full stepbystep solution to problem in Discrete Mathematics with Applications were answered by , our top Math solution expert on 07/19/17, 06:34AM. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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