 Chapter 1: The Foundations: Logic and Proofs
 Chapter 1.1: The Foundations: Logic and Proofs
 Chapter 1.2: The Foundations: Logic and Proofs
 Chapter 1.3: The Foundations: Logic and Proofs
 Chapter 1.4: The Foundations: Logic and Proofs
 Chapter 1.5: The Foundations: Logic and Proofs
 Chapter 1.6: The Foundations: Logic and Proofs
 Chapter 1.7: The Foundations: Logic and Proofs
 Chapter 10.1: Trees
 Chapter 10.2: Trees
 Chapter 10.3: Trees
 Chapter 11: Boolean Algebra
 Chapter 11.1: Boolean Algebra
 Chapter 11.2: Boolean Algebra
 Chapter 11.3: Boolean Algebra
 Chapter 11.4: Boolean Algebra
 Chapter 12: Modeling Computation
 Chapter 12.1: Modeling Computation
 Chapter 12.2: Modeling Computation
 Chapter 12.3: Modeling Computation
 Chapter 12.4: Modeling Computation
 Chapter 12.5: Modeling Computation
 Chapter 2: Basic Structures: Sets, Functions, Sequences, and Sums
 Chapter 2.1: Basic Structures: Sets, Functions, Sequences, and Sums
 Chapter 2.2: Basic Structures: Sets, Functions, Sequences, and Sums
 Chapter 2.3: Basic Structures: Sets, Functions, Sequences, and Sums
 Chapter 2.4: Basic Structures: Sets, Functions, Sequences, and Sums
 Chapter 3: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.1: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.2: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.3: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.4: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.5: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.6: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.7: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 3.8: The Fundamentals: Algorithms, the Integers, and Matrices
 Chapter 4: Induction and Recursion
 Chapter 4.1: Induction and Recursion
 Chapter 4.2: Induction and Recursion
 Chapter 4.3: Induction and Recursion
 Chapter 4.4: Induction and Recursion
 Chapter 4.5: Induction and Recursion
 Chapter 5: Counting
 Chapter 5.1: Counting
 Chapter 5.2: Counting
 Chapter 5.3: Counting
 Chapter 5.4: Counting
 Chapter 5.5: Counting
 Chapter 5.6: Counting
 Chapter 6: Discrete Probability
 Chapter 6.1: Discrete Probability
 Chapter 6.2: Discrete Probability
 Chapter 6.3: Discrete Probability
 Chapter 6.4: Discrete Probability
 Chapter 7: Advanced Counting Techniques
 Chapter 7.1: Advanced Counting Techniques
 Chapter 7.2: Advanced Counting Techniques
 Chapter 7.3: Advanced Counting Techniques
 Chapter 7.4: Advanced Counting Techniques
 Chapter 7.5: Advanced Counting Techniques
 Chapter 7.6: Advanced Counting Techniques
 Chapter 8: Relations
 Chapter 8.1: Relations
 Chapter 8.2: Relations
 Chapter 8.3: Relations
 Chapter 8.4: Relations
 Chapter 8.5: Relations
 Chapter 8.6: Relations
 Chapter 9: Graphs
 Chapter 9.1: Graphs
 Chapter 9.2: Graphs
 Chapter 9.3: Graphs
 Chapter 9.4: Graphs
 Chapter 9.5: Graphs
 Chapter 9.6: Graphs
 Chapter 9.7: Graphs
 Chapter 9.8: Graphs
 Chapter A1: Axioms for the Real Numbers and the Positive Integers
 Chapter A2: Exponential and Logarithmic Functions
 Chapter A3: Pseudocode
Discrete Mathematics and Its Applications 6th Edition  Solutions by Chapter
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Discrete Mathematics and Its Applications  6th Edition  Solutions by Chapter
Get Full SolutionsDiscrete Mathematics and Its Applications was written by Patricia and is associated to the ISBN: 9780073229720. The full stepbystep solution to problem in Discrete Mathematics and Its Applications were answered by Patricia, our top Math solution expert on 01/16/18, 07:40PM. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 6. Since problems from 80 chapters in Discrete Mathematics and Its Applications have been answered, more than 6849 students have viewed full stepbystep answer. This expansive textbook survival guide covers the following chapters: 80.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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!

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

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

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Solvable system Ax = b.
The right side b is in the column space of A.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
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