 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 44044 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. 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.

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

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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Companion matrix.
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 nl  An).

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.

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).

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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

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

Solvable system Ax = b.
The right side b is in the column space of A.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

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