 Chapter 10.1: Let P D .f1; 2; 3; : : : ; 20g; j/; that is, P is the poset whose e...
 Chapter 10.2: Let C be a chain and A be an antichain of a poset P D .X; /. Prove ...
 Chapter 10.3: Let P D .X; / be a poset. Suppose there are chains C1 and C2 in P s...
 Chapter 10.4: Let P D .X; / be a poset. Prove that P is an antichain if and only ...
 Chapter 10.5: Let P D .X; / be a finite poset. We say that P is a weak order if w...
 Chapter 10.6: Let P D .X; / be a poset. We say that P is a semiorder if we can as...
 Chapter 10.7: to each element x 2 X a real interval ax; bx such that x < y in P i...
 Chapter 10.8: Let P be the poset whose Hasse diagram is shown in the figure. How ...
 Chapter 10.9: Let P be the poset whose Hasse diagram is shown in the figure. a b ...
 Chapter 10.10: c. Prove that there can be no linear extension of P in which f < a ...
 Chapter 10.11: Let P D .X; / be a lattice, and suppose that for all x; y 2 X, we h...
 Chapter 10.12: Recall from Definition 54.5 and Example 54.6 that the set of all pa...
 Chapter 10.13: Let P D .X; / be a lattice. Let a; x1; x2; : : : ; xn 2 X and suppo...
 Chapter 10.14: Let P D .X; / be a finite poset. Let a; b 2 X and define U.a; b/ D ...
Solutions for Chapter Chapter 10: Partially Ordered Sets
Full solutions for Mathematics: A Discrete Introduction  3rd Edition
ISBN: 9780840049421
Solutions for Chapter Chapter 10: Partially Ordered Sets
Get Full SolutionsThis textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Since 14 problems in chapter Chapter 10: Partially Ordered Sets have been answered, more than 9020 students have viewed full stepbystep solutions from this chapter. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421. Chapter Chapter 10: Partially Ordered Sets includes 14 full stepbystep solutions.

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).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Elimination.
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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

lAII = 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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.