- 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
cond(A) = c(A) = IIAIlIIA-III = 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.
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
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or firstname.lastname@example.org
Forgot password? Reset it here