 1.8.26E: Suppose that five ones and four zeros are arranged around a circle....
 1.8.17E: Suppose that a and b are odd integers with a ? b. Show there is a u...
 1.8.18E: Show that if r is an irrational number, there is a unique integer n...
 1.8.19E: Show that if n is an odd integer, then there is a unique integer k ...
 1.8.20E: Prove that given a real number x there exist unique numbers n and ?...
 1.8.29E: Prove that there is no positive integer n such that n2 + n3 = 100.
 1.8.21E: Prove that given a real number x there exist unique numbers n and ?...
 1.8.30E: Prove that there are no solutions in integers x and y to the equati...
 1.8.22E: Use forward reasoning to show that if x is a nonzero real number, t...
 1.8.31E: Prove that there are no solutions in positive integers x and y to t...
 1.8.23E: The harmonic mean of two real numbers x and y equals 2xy/(x + y). B...
 1.8.24E: The quadratic mean of two real numbers x and y equals By computing ...
 1.8.25E: Write the numbers 1,2,...,2n on a blackboard, where n is an odd int...
 1.8.32E: Prove that there are infinitely many solutions in positive integers...
 1.8.33E: Adapt the proof in Example 4 in Section 1.7 to prove that if n = ab...
 1.8.34E: Prove that is irrational.
 1.8.28E: Formulate a conjecture about the final two decimal digits of the sq...
 1.8.35E: Prove that between every two rational numbers there is an irrationa...
 1.8.36E: Prove that between every rational number and every irrational numbe...
 1.8.37E: Let S =x1y1+ x2y2 +……..+ xnyn,Where x1,x2………. x nand y1,y2………. yn a...
 1.8.1E: Prove that n2 + I ? 2n when n is a positive integer with 1 ? n ? 4.
 1.8.2E: Prove that there are no positive perfect cubes less than 1000 that ...
 1.8.3E: Prove that if x and y are real numbers, then max(x, y) + min (x, y)...
 1.8.39E: Verify the 3x + 1 conjecture for these integers. a) 6 b) 7 c) 17 d) 21
 1.8.40E: Verify the 3x + 1 conjecture for these integers. a) 16 b) 11 c) 35 ...
 1.8.10E: Prove that either 210500 + 15 or 2 ? 10500+ 16 is not a perfect squ...
 1.8.4E: Use a proof by cases to show that min (a, min(b, c)) = min(min(a, b...
 1.8.5E: Prove using the notion of without loss of generality that min( x , ...
 1.8.6E: Prove using the notion of without loss of generality that 5x + 5y i...
 1.8.42E: Prove or disprove that you can use dominoes to tile a standard chec...
 1.8.41E: Prove or disprove that you can use dominoes to tile the standard ch...
 1.8.43E: Prove that you can use dominoes to tile a rectangular checkerboard ...
 1.8.11E: Prove that there exists a pair of consecutive integers such that on...
 1.8.13E: Prove or disprove that there is a rational number x and an irration...
 1.8.47E: Show that by removing two white squares and two black squares from ...
 1.8.7E: Prove the triangle inequality, which stales that if x and y are rea...
 1.8.8E: Prove that there is a positive integer that equals the sum of the p...
 1.8.9E: Prove that there are 100 consecutive positive integers that are not...
 1.8.44E: Prove or disprove that you can use dominoes to tile a 5x5 checkerbo...
 1.8.45E: Use a proof by exhaustion to show that a tiling using dominoes of a...
 1.8.46E: Prove that when a while square and a black square are removed from ...
 1.8.48E: Find all squares. if they exist, on an 8×8 checkerboard such that t...
 1.8.49E: a) Draw each of the five different tetrominoes, where a tetromino i...
 1.8.38E: Prove or disprove that if you have an 8gallon jug of water and two...
 1.8.27E: Formulate a conjecture about the decimal digits that appear as the ...
 1.8.14E: Prove or disprove that if a and b are rational numbers, then ab is ...
 1.8.15E: Show that each of these statements can be used to express the fact ...
 1.8.50E: Prove or disprove that you can tile a 10 ×10 checkerboard using str...
 1.8.16E: Show that if a, b, and c are real numbers and a ? 0, then there is ...
Solutions for Chapter 1.8: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 1.8
Get Full SolutionsSince 49 problems in chapter 1.8 have been answered, more than 125959 students have viewed full stepbystep solutions from this chapter. Chapter 1.8 includes 49 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. This expansive textbook survival guide covers the following chapters and their solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

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.

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

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

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

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

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 .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

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.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.