 1.3.1E: Let A = {2, 3, 4} and B = {6, 8, 10} and define a relation R from A...
 1.3.2E: Let C = D = {?3,?2,?1, 1, 2, 3} and define a relation S from C to D...
 1.3.3E: Let E = {1, 2, 3} and F = {–2, 1,0} and define a relation T from E...
 1.3.4E: Let G = {–2, 0, 2} and H = {4, 6, 8} and define a relation V from G...
 1.3.5E: Define a relation S from R to R as follows:For all (x, y)? R × R,(x...
 1.3.6E: Define a relation R from R to R as follows:For all ( x , y) ? R × R...
 1.3.7E: Let A = {4, 5, 6} and B = {5, 6, 7} and define relations R, S, and ...
 1.3.8E: Let A = {2, 4} and B = {1, 3, 5} and define relations U, V, and W f...
 1.3.9E: a. Find all relations from {0,1} to {1}.b. Find all functions from ...
 1.3.10E: Find four relations from {a, b} to {x, y} that are not functions fr...
 1.3.11E: Define a relation P from R+ to R as follows: For all real numbers x...
 1.3.12E: Define a relation T from R to R as follows: For all real numbers x ...
 1.3.13E: Let A = {–1, 0, 1} and B = {t, u, v, w}. Define a function F: A ? B...
 1.3.14E: Let C = {1, 2, 3, 4} and D = {a, b, c, d}. Define a function G: C ?...
 1.3.15E: Let X = {2, 4, 5} and Y = {1, 2, 4, 6}. Which of the following arro...
 1.3.16E: Let f be the squaring function defined in Example. Find f (–1), f (...
 1.3.17E: Let g be the successor function defined in Example. Find g(–1000), ...
 1.3.18E: Let h be the constant function defined in Example. Find ExampleFunc...
 1.3.19E: Define functions f and g from R to R by the following formulas: For...
 1.3.20E: Define functions H and K from R to R by the following formulas: For...
Solutions for Chapter 1.3: Discrete Mathematics with Applications 4th Edition
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 1.3
Get Full SolutionsDiscrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.3 includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Since 20 problems in chapter 1.3 have been answered, more than 49084 students have viewed full stepbystep solutions from this chapter.

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

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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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 II· Condition numbers measure the sensitivity of the output to change in the input.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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].

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

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.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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