- Chapter 1: Joy
- Chapter 10: Sets I: Introduction, Subsets
- Chapter 11: Sets I: Introduction, Subsets
- Chapter 12: Sets II: Operations
- Chapter 13: Combinatorial Proof: Two Examples
- Chapter 14: Relations
- Chapter 15: Equivalence Relations
- Chapter 16: Partitions
- Chapter 17: Binomial Coefficients
- Chapter 18: Counting Multisets
- Chapter 19: Inclusion-Exclusion
- Chapter 2: Speaking (and Writing) of Mathematics
- Chapter 20: Contradiction
- Chapter 21: Smallest Counterexample
- Chapter 22: Induction
- Chapter 23: Recurrence Relations
- Chapter 24: Functions
- Chapter 25: The Pigeonhole Principle
- Chapter 26: Composition
- Chapter 27: Permutations
- Chapter 28: Symmetry
- Chapter 29: Assorted Notation
- Chapter 3: Definition
- Chapter 30: Sample Space
- Chapter 31: Events
- Chapter 32: Conditional Probability and Independence
- Chapter 33: Random Variables
- Chapter 34: Expectation
- Chapter 35: Dividing
- Chapter 36: Greatest Common Divisor
- Chapter 37: Modular Arithmetic
- Chapter 38: The Chinese Remainder Theorem
- Chapter 39: Factoring
- Chapter 4: Theorem
- Chapter 40: Groups
- Chapter 41: Group Isomorphism The Same?
- Chapter 42: Subgroups
- Chapter 43: Fermats Little Theorem
- Chapter 44: Public Key Cryptography I: Introduction The Problem: Private Communication in Public
- Chapter 45: Public Key Cryptography II: Rabins Method
- Chapter 46: Public Key Cryptography III: RSA
- Chapter 47: Fundamentals of Graph Theory
- Chapter 48: Subgraphs
- Chapter 49: Connection
- Chapter 5: Proof
- Chapter 50: Trees
- Chapter 51: Eulerian Graphs
- Chapter 52: Coloring
- Chapter 53: Planar Graphs
- Chapter 54: Fundamentals of Partially Ordered Sets
- Chapter 55: Max and Min
- Chapter 56: Linear Orders
- Chapter 57: Linear Extensions
- Chapter 58: Dimension
- Chapter 59: Lattices
- Chapter 6: Counterexample
- Chapter 7: Boolean Algebra
- Chapter 8: Lists
- Chapter 9: Factorial
- Chapter Chapter 1: Fundamentals
- Chapter Chapter 10: Partially Ordered Sets
- Chapter Chapter 2: Collections
- Chapter Chapter 3: Counting and Relations
- Chapter Chapter 4: More Proof
- Chapter Chapter 5: Functions
- Chapter Chapter 6: Probability
- Chapter Chapter 7: Number Theory
- Chapter Chapter 8: Algebra
- Chapter Chapter 9: Graphs
Mathematics: A Discrete Introduction 3rd Edition - Solutions by Chapter
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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].
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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 .
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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