 Chapter 1.1: Statements, Symbolic Representation, and Tautologies
 Chapter 1.2: Propositional Logic
 Chapter 1.3: Quantifiers, Predicates, and Validity
 Chapter 1.4: Predicate Logic
 Chapter 1.5: Logic Programming
 Chapter 1.6: Logic Programming
 Chapter 2.1: Proof Techniques
 Chapter 2.2: Induction
 Chapter 2.3: More on Proof of Correctness
 Chapter 2.4: Number Theory
 Chapter 3.1: Recursive Definitions
 Chapter 3.2: Recurrence Relations
 Chapter 3.3: Analysis of Algorithms
 Chapter 4.1: Sets
 Chapter 4.2: Counting
 Chapter 4.3: Principle of Inclusion and Exclusion; Pigeonhole Principle
 Chapter 4.4: Permutations and Combinations
 Chapter 5.1: Relations
 Chapter 5.2: Topological Sorting
 Chapter 5.3: Relations and Databases
 Chapter 5.4: Functions
 Chapter 5.5: Order of Magnitude
 Chapter 5.6: The Mighty Mod Function
 Chapter 5.7: Matrices
 Chapter 6.1: Graphs and Their Representations
 Chapter 6.2: Trees and Their Representations
 Chapter 6.3: Decision Trees
 Chapter 6.4: Huffman Codes
 Chapter 7.1: Directed Graphs and Binary Relations; Warshalls Algorithm
 Chapter 7.2: Euler Path and Hamiltonian Circuit
 Chapter 7.3: Shortest Path and Minimal Spanning Tree
 Chapter 7.4: Traversal Algorithms
 Chapter 7.5: Articulation Points and Computer Networks
 Chapter 8.1: Boolean Algebra Structure
 Chapter 8.2: Logic Networks
 Chapter 8.3: Minimization
 Chapter 9.1: Algebraic Structures
 Chapter 9.2: Coding Theory
 Chapter 9.3: FiniteState Machines
 Chapter 9.4: Turing Machines
 Chapter 9.5: Formal Languages
Mathematical Structures for Computer Science 7th Edition  Solutions by Chapter
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Mathematical Structures for Computer Science  7th Edition  Solutions by Chapter
Get Full SolutionsThe full stepbystep solution to problem in Mathematical Structures for Computer Science were answered by , our top Math solution expert on 01/18/18, 05:04PM. Since problems from 41 chapters in Mathematical Structures for Computer Science have been answered, more than 16301 students have viewed full stepbystep answer. Mathematical Structures for Computer Science was written by and is associated to the ISBN: 9781429215107. This expansive textbook survival guide covers the following chapters: 41. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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

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

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.