- Chapter 1.1:
- Chapter 1.1: Variables
- Chapter 1.2:
- Chapter 1.2: The Language of Sets
- Chapter 1.3:
- Chapter 1.3: The Language of Relations and Functions
- Chapter 10.1:
- Chapter 10.1: Graphs: Definitions and Basic Properties
- Chapter 10.2:
- Chapter 10.2: Trails, Paths, and Circuits
- Chapter 10.3:
- Chapter 10.3: Matrix Representations of Graphs
- Chapter 10.4:
- Chapter 10.4: Isomorphisms of Graphs
- Chapter 10.5:
- Chapter 10.5: Trees
- Chapter 10.6:
- Chapter 10.6: Rooted Trees
- Chapter 10.7:
- Chapter 10.7: Spanning Trees and Shortest Paths
- Chapter 11.1:
- Chapter 11.1: Real-Valued Functions of a Real Variable and Their Graphs
- Chapter 11.2:
- Chapter 11.2: O-, -, and -Notations
- Chapter 11.3:
- Chapter 11.3: Application: Analysis of Algorithm Efficiency I
- Chapter 11.4:
- Chapter 11.4: Exponential and Logarithmic Functions: Graphs and Orders
- Chapter 11.5:
- Chapter 11.5: Application: Analysis of Algorithm Efficiency II
- Chapter 12.1:
- Chapter 12.1: Formal Languages and Regular Expressions
- Chapter 12.2:
- Chapter 12.2: Finite-State Automata
- Chapter 12.3:
- Chapter 12.3: Simplifying Finite-State Automata
- Chapter 2.1:
- Chapter 2.1: Logical Form and Logical Equivalence
- Chapter 2.2:
- Chapter 2.2: Conditional Statements
- Chapter 2.3:
- Chapter 2.3: Valid and Invalid Arguments
- Chapter 2.4:
- Chapter 2.4: Application: Digital Logic Circuits
- Chapter 2.5:
- Chapter 2.5: Application: Number Systems and Circuits for Addition
- Chapter 3.1: Predicates and Quantified Statements I
- Chapter 3.2:
- Chapter 3.2: Predicates and Quantified Statements II
- Chapter 3.3:
- Chapter 3.3: Statements with Multiple Quantifiers
- Chapter 3.4:
- Chapter 3.4: Arguments with Quantified Statements
- Chapter 4.1:
- Chapter 4.1: Direct Proof and Counterexample I: Introduction
- Chapter 4.2:
- Chapter 4.2: Direct Proof and Counterexample II: Rational Numbers
- Chapter 4.3:
- Chapter 4.3: Direct Proof and Counterexample III: Divisibility
- Chapter 4.4:
- Chapter 4.4: Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem
- Chapter 4.5:
- Chapter 4.5: Direct Proof and Counterexample V: Floor and Ceiling
- Chapter 4.6:
- Chapter 4.6: Indirect Argument: Contradiction and Contraposition
- Chapter 4.7:
- Chapter 4.7: Indirect Argument: Two Classical Theorems
- Chapter 4.8:
- Chapter 4.8: Application: Algorithms
- Chapter 5.1:
- Chapter 5.1: Sequences
- Chapter 5.2:
- Chapter 5.2: Mathematical Induction I
- Chapter 5.3:
- Chapter 5.3: Mathematical Induction II
- Chapter 5.4:
- Chapter 5.4: Strong Mathematical Induction and the Well-Ordering Principle for the Integers
- Chapter 5.5:
- Chapter 5.5: Application: Correctness of Algorithms
- Chapter 5.6:
- Chapter 5.6: Defining Sequences Recursively
- Chapter 5.7:
- Chapter 5.7: Solving Recurrence Relations by Iteration
- Chapter 5.8:
- Chapter 5.8: Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients
- Chapter 5.9:
- Chapter 5.9: General Recursive Definitions and Structural Induction
- Chapter 6.1:
- Chapter 6.1: Set Theory: Definitions and the Element Method of Proof
- Chapter 6.2:
- Chapter 6.2: Properties of Sets
- Chapter 6.3:
- Chapter 6.3: Disproofs, Algebraic Proofs, and Boolean Algebras
- Chapter 6.4:
- Chapter 6.4: Boolean Algebras, Russells Paradox, and the Halting Problem
- Chapter 7.1:
- Chapter 7.1: Functions Defined on General Sets
- Chapter 7.2:
- Chapter 7.2: One-to-One and Onto, Inverse Functions
- Chapter 7.3:
- Chapter 7.3: Composition of Functions
- Chapter 7.4:
- Chapter 7.4: Cardinality with Applications to Computability
- Chapter 8.1:
- Chapter 8.1: Relations on Sets
- Chapter 8.2:
- Chapter 8.2: Reflexivity, Symmetry, and Transitivity
- Chapter 8.3:
- Chapter 8.3: Equivalence Relations
- Chapter 8.4:
- Chapter 8.4: Modular Arithmetic with Applications to Cryptography
- Chapter 8.5:
- Chapter 8.5: Partial Order Relations
- Chapter 9.1:
- Chapter 9.1: Introduction
- Chapter 9.2:
- Chapter 9.2: Possibility Trees and the Multiplication Rule
- Chapter 9.3:
- Chapter 9.3: Counting Elements of Disjoint Sets: The Addition Rule
- Chapter 9.4:
- Chapter 9.4: The Pigeonhole Principle
- Chapter 9.5:
- Chapter 9.5: Counting Subsets of a Set: Combinations
- Chapter 9.6:
- Chapter 9.6: r-Combinations with Repetition Allowed
- Chapter 9.7:
- Chapter 9.7: Pascals Formula and the Binomial Theorem
- Chapter 9.8:
- Chapter 9.8: Probability Axioms and Expected Value
- Chapter 9.9:
- Chapter 9.9: Conditional Probability, Bayes Formula, and Independent Events
Discrete Mathematics with Applications 4th Edition - Solutions by Chapter
Full solutions for Discrete Mathematics with Applications | 4th Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
peA) = det(A - AI) has peA) = zero matrix.
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].
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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.
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.
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 ].
Solvable system Ax = b.
The right side b is in the column space of A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).