 1.3.1: Let A = {2, 3, 4} and B = {6, 8, 10} and define a relation R from A...
 1.3.2: Let C = D = {3, 2, 1, 1, 2, 3} and define a relation S from C to D ...
 1.3.3: Let E = {1, 2, 3} and F = {2, 1, 0} and define a relation T from E ...
 1.3.4: Let G = {2, 0, 2} and H = {4, 6, 8} and define a relation V from G ...
 1.3.5: Let G = {2, 0, 2} and H = {4, 6, 8} and define a relation V from G ...
 1.3.6: Define a relation R from R to R as follows: For all (x, y) R R, (x,...
 1.3.7: Let A = {4, 5, 6} and B = {5, 6, 7} and define relations R, S, and ...
 1.3.8: Let A = {2, 4} and B = {1, 3, 5} and define relations U, V, and W f...
 1.3.9: a. Find all relations from {0,1} to {1}. b. Find all functions from...
 1.3.10: Find four relations from {a, b} to {x, y} that are not functions fr...
 1.3.11: Define a relation P from R+ to R as follows: For all real numbers x...
 1.3.12: Define a relation T from R to R as follows: For all real numbers x ...
 1.3.13: Let A = {1, 0, 1} and B = {t, u, v, w}. Define a function F: A B by...
 1.3.14: Let C = {1, 2, 3, 4} and D = {a, b, c, d}. Define a function G: C D...
 1.3.15: Let X = {2, 4, 5} and Y = {1, 2, 4, 6}. Which of the following arro...
 1.3.16: Let f be the squaring function defined in Example 1.3.6. Find f (1)...
 1.3.17: Let g be the successor function defined in Example 1.3.6. Find g(10...
 1.3.18: Let h be the constant function defined in Example 1.3.6. Find h 12 ...
 1.3.19: Define functions f and g from R to R by the following formulas: For...
 1.3.20: Define functions H and K from R to R by the following formulas: For...
Solutions for Chapter 1.3: The Language of Relations and Functions
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 1.3: The Language of Relations and Functions
Get Full SolutionsSince 20 problems in chapter 1.3: The Language of Relations and Functions have been answered, more than 23942 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4th. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.3: The Language of Relations and Functions includes 20 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).

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Outer product uv T
= column times row = rank one matrix.

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.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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