- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
A symmetric matrix with eigenvalues of both signs (+ and - ).
A sequence of steps intended to approach the desired solution.
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.
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.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).