 2.SE.1E: Let A be the set of English words that contain the letter x, and le...
 2.SE.2E: Show that if A is a subset of B. then the power set of, A is a subs...
 2.SE.3E: Suppose that, A and B are sets such that the power set of A is a su...
 2.SE.4E: Let E denote the set of even integers and O denote the set of odd i...
 2.SE.5E: Show that if A and B are sets, then A ? (A ? B) = A ? B.
 2.SE.6E: Let A and B be sets. Show that A ? B if and only if A ? B = A.
 2.SE.7E: Let A, B, and C be sets. Show that (A ? B) ? C is not necessarily e...
 2.SE.8E: Suppose that A, B, and C are sets. Prove or disprove that (A ? B)?C...
 2.SE.9E: Suppose that A, B, C. and D are sets. Prove or disprove that (A ? B...
 2.SE.10E: Show that if A and B are finite sets, then A ? B ?  A ? B. Dete...
 2.SE.11E: Let A and B be sets in a finite universal set U. List the following...
 2.SE.12E: Let A and B be subsets of the finite universal set U. Show that
 2.SE.13E: Let f and g be functions from (1,. 2,. 3,. 4) to (a, b, c, d) and f...
 2.SE.14E: Suppose that f is a function from A to B where A and B are finite s...
 2.SE.15E: Suppose that is a function from A to B where A and If are finite se...
 2.SE.16E: Suppose that f is a function from the set A to the set B. Prove tha...
 2.SE.17E: Prove that if f and g are functions from A to B and Sf = Sg (using ...
 2.SE.18E: Show that if n is an integer, then n = ?n/2? + ?n/2?.
 2.SE.19E: For which real numbers x and y is it true that ?x +y? = ? x? +? y ?
 2.SE.20E: For which real numbers x and y is it true that ?x +y? = ? x? +? y ?
 2.SE.21E: For which real numbers x and y is it true that ?x +y? = ? x? +? y ?
 2.SE.22E: Prove that ?n/2? ? n/2? = ?n2/4? for all integers n.
 2.SE.23E: Prove that if m is an integer, then ?x? +?m ? x? = m ? 1unless x is...
 2.SE.24E: Prove that if x is a real number, then ??x/2? /2 ? = ? x/4 ?.
 2.SE.25E: Prove that if n is an odd integer, then ?n2/4? = (n2+ 3)/4.
 2.SE.26E: ?Prove that if \(m\) and \(n\) are positive integers and \(x\) is a...
 2.SE.27E: ?Prove that if \(m\) is a positive integer and \(x\) is a real numb...
 2.SE.28E: We define the Ulam numbers by setting u1= 1 and u2 = 2. Furthermore...
 2.SE.29E: ?Determine the value of \(\prod_{k=1}^{100} \frac{k+1}{k}\). (The n...
 2.SE.30E: Determine a rule for generating the terms of the sequence that begi...
 2.SE.31E: Determine a rule for generating the terms of the sequence that begi...
 2.SE.32E: Show that the set of irrational numbers is an uncountable set.
 2.SE.33E: Show that the set S is a countable set if there is a function f fro...
 2.SE.34E: Show that the set of all finite subsets of the set of positive inte...
 2.SE.35E: Show that R × R = R. [Hint: Use the Schröder Bernstein theorem...
 2.SE.36E: Show that C. the set of complex numbers has the same cardinality as...
 2.SE.37E: Find A" if A is
 2.SE.38E: Show that if A = cI, where c is a real number and I is the n × n id...
 2.SE.39E: Show that if A is a 2 × 2 matrix such that AB = BA whenever B is a ...
 2.SE.40E: Show that if A and B are invertible matrices and AB exists, then (A...
 2.SE.41E: Let A be an 11 × 11 matrix and let 0 be the 11 × 11 matrix all of w...
Solutions for Chapter 2.SE: Sequences and Summations
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 2.SE: Sequences and Summations
Get Full SolutionsSummary of Chapter 2.SE: Sequences and Summations
Sequences are ordered lists of elements, used in discrete mathematics in many ways. For example, they can be used to represent solutions to certain counting problems
This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.SE: Sequences and Summations includes 41 full stepbystep solutions. Since 41 problems in chapter 2.SE: Sequences and Summations have been answered, more than 698464 students have viewed full stepbystep solutions from this chapter. 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.

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).

Column space C (A) =
space of all combinations of the columns of A.

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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.