 5.2.1: A relation R is defined on N N by (a, b) R (c, d) if a + d = b + c....
 5.2.2: Let S = {1, 2, 3, 4, 5, 6, 7}. The relation R = {(1, 1), (1, 3), (1...
 5.2.3: Let R be an equivalence relation on the set S = {a, b, c, d, e, f}....
 5.2.4: Let R be an equivalence relation on the set S = {u, v,w, x, y, z} h...
 5.2.5: An equivalence relation R on the set S = {1, 2, 3, 4, 5, 6} results...
 5.2.6: Let R denote the set of nonzero real numbers. Define a relation R o...
 5.2.7: Let R be a relation defined on Z by a R b if a + b = 0 or a b = 0. ...
 5.2.8: A relation R is defined on the set S = {7,5,4,1, 3, 4, 9} by a R b ...
 5.2.9: A relation R is defined on the set Z of integers by a R b if 11a 5b...
 5.2.10: A relation R is defined on Z Z by (a, b) R (c, d) if a + b + c + d ...
 5.2.11: A relation R is defined on ZZ by (a, b) R (c, d) if abcd is even. I...
 5.2.12: Let S = {x, y, z}. A relation R on S has the following four propert...
 5.2.13: Does there exist an example of an equivalence relation R on the set...
 5.2.14: A relation R is defined on the set R+ of positive real numbers by a...
 5.2.15: Let S be a nonempty set and let P = {S1, S2, . . . , Sk} be a parti...
 5.2.16: Give an example of an equivalence relation R on the set A = {1, 2, ...
 5.2.17: Prove or disprove: Let A be a finite set with A = n 2. There exis...
Solutions for Chapter 5.2: Equivalence Relations
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 5.2: Equivalence Relations
Get Full SolutionsThis textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Chapter 5.2: Equivalence Relations includes 17 full stepbystep solutions. Since 17 problems in chapter 5.2: Equivalence Relations have been answered, more than 12945 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. 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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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.

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.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).