- 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: Finite-State 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
Mathematical Structures for Computer Science | 7th Edition - Solutions by ChapterGet Full Solutions
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 .
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.
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.
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).
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.