- 2.R.1RQ: Explain what it means for one set to be a subset of another set. Ho...
- 2.R.2RQ: What is the empty set ? Show that the empty set is a subset of ever...
- 2.R.3RQ: a) Define | S |, the cardinality of the set S.________________b) Gi...
- 2.R.4RQ: a) Define the power set of a set S.________________b) When is the e...
- 2.R.5RQ: a) Define the union, intersection, difference, and symmetric differ...
- 2.R.6RQ: a) Explain what it means for two sets to be equal.________________b...
- 2.R.7RQ: Explain the relationship between logical equivalences and set ident...
- 2.R.8RQ: a) Define the domain, codomain, and range of a function.___________...
- 2.R.9RQ: a) Define what it means for a function from the set of positive int...
- 2.R.10RQ: a) Define the inverse of a function.________________b) When docs a ...
- 2.R.11RQ: a) Define the floor and ceiling functions from the set of real numb...
- 2.R.12RQ: Conjecture a formula for the terms of the sequence that begins 8, 1...
- 2.R.13RQ: Suppose that an = an?1 ?5 for n = 1, 2, .... Find a formula for an.
- 2.R.14RQ: What is the sum of the terms of the geometric progression a + ar +....
- 2.R.15RQ: Show that the set of odd integers is countable.
- 2.R.16RQ: Give an example of an uncountable set.
- 2.R.17RQ: Define the product of two matrices A and B. When is this product de...
- 2.R.18RQ: Show that matrix multiplication is not commutative.
Solutions for Chapter 2.R: Functions
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
ISBN: 9780073383095
Solutions for Chapter 2.R
11
3
Summary of Chapter 2.R: Functions
The concept of a function is extremely important in mathematics and computer science. For example, in discrete mathematics functions are used in the definition of such discrete structures as sequences and strings.
Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Since 18 problems in chapter 2.R: Functions have been answered, more than 732802 students have viewed full step-by-step solutions from this chapter. Chapter 2.R: Functions includes 18 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions.
Key Math Terms and definitions covered in this textbook
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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.
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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).
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Diagonalization
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
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Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.
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Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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lA-II = 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.
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Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
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Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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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).
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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.
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Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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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.
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
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Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).