 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 Patricia, 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 3518 students have viewed full stepbystep answer. Mathematical Structures for Computer Science was written by Patricia 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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

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

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).

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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 v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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