- 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
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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 n-l - An).
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].
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.
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
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.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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.
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
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.
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
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.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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.
Having trouble accessing your account? Let us help you, contact support at +1(510) 944-1054 or email@example.com
Forgot password? Reset it here