 5.5.1: Disprove the statement: If a and b are any two real numbers, then l...
 5.5.2: Disprove the statement: If n {0, 1, 2, 3, 4}, then 2n + 3n + n(n 1)...
 5.5.3: Disprove the statement: If n {1, 2, 3, 4, 5}, then 3  (2n2 + 1).
 5.5.4: Disprove the statement: Let n N. If n(n+1)2 is odd, then (n+1)(n+2)...
 5.5.5: Disprove the statement: For every two positive integers a and b, (a...
 5.5.6: Let a, b Z. Disprove the statement: If ab and (a + b)2 are of oppos...
 5.5.7: For positive real numbers a and b, it can be shown that (a + b) 1a ...
 5.5.8: In Exercise 5.7, it is stated that (a + b) 1a + 1b 4 for every two ...
 5.5.9: Disprove the statement: For every positive integer x and every inte...
 5.5.10: Prove that there is no largest negative rational number.
 5.5.11: Prove that there is no smallest positive irrational number
 5.5.12: Prove that 200 cannot be written as the sum of an odd integer and t...
 5.5.13: Use proof by contradiction to prove that if a and b are odd integer...
 5.5.14: Prove that if a 2 and b are integers, then a  b or a  (b + 1).
 5.5.15: Prove that 1000 cannot be written as the sum of three integers, an ...
 5.5.16: Prove that the product of an irrational number and a nonzero ration...
 5.5.17: Prove that when an irrational number is divided by a (nonzero) rati...
 5.5.18: Let a be an irrational number and r a nonzero rational number. Prov...
 5.5.19: Prove that 3 is irrational. [Hint: First prove for an integer a tha...
 5.5.20: Prove that 2 + 3 is an irrational number.
 5.5.21: (a) Prove that 6 is an irrational number.(b) Prove that there are i...
 5.5.22: Let S = {p + q 2 : p, q Q} and T = {r + s 3 : r,s Q}. Prove that S ...
 5.5.23: Prove that there is no integer a such that a 5 (mod 14) and a 3 (mo...
 5.5.24: Prove that there exists no positive integer x such that 2x < x 2 < 3x
 5.5.25: Prove that there do not exist three distinct positive integers a, b...
 5.5.26: Prove that the sum of the squares of two odd integers cannot be the...
 5.5.27: Prove that if x and y are positive real numbers, then x + y = x + y.
 5.5.28: Prove that there do not exist positive integers m and n such that m...
 5.5.29: Let m be a positive integer of the form m = 2s, where s is an odd i...
 5.5.30: Prove that there do not exist three distinct real numbers a, b and ...
 5.5.31: Use a proof by contradiction to prove the following. Let m Z. If 3 ...
 5.5.32: Prove that there exist no positive integers m and n for which m2 + ...
 5.5.33: (a) Prove that there is no rational number solution of the equation...
 5.5.34: Prove that if n is an odd integer, then 7n 5 is even by(a) a direct...
 5.5.35: Let x be a positive real number. Prove that if x 2x > 1, then x > 2...
 5.5.36: Let a, b R. Prove that if ab = 0, then a = 0 by using as many of th...
 5.5.37: Let x, y R+. Prove that if x y, then x 2 y2 by(a) a direct proof, (...
 5.5.38: Prove the following statement using more than one method of proof.L...
 5.5.39: Prove the following statement using more than one method of proof.F...
 5.5.40: Show that there exist a rational number a and an irrational number ...
 5.5.41: Show that there exist a rational number a and an irrational number ...
 5.5.42: Show that there exist two distinct irrational numbers a and b such ...
 5.5.43: Show that there exist no nonzero real numbers a and b such that a2 ...
 5.5.44: Prove that there exists a unique real number solution to the equati...
 5.5.45: Let R(x) be an open sentence over a domain S. Suppose that x S, R(x...
 5.5.46: (a) Prove that there exist four distinct positive integers such tha...
 5.5.47: Let S be a set of three integers. For a nonempty subset A of S, let...
 5.5.48: Prove that the equation cos2(x) 4x + = 0 has a real number solution...
 5.5.49: Disprove the statement: There exist odd integers a and b such that ...
 5.5.50: Disprove the statement: There is a real number x such that x 6 + x ...
 5.5.51: Disprove the statement: There is an integer n such that n4 + n3 + n...
 5.5.52: The integers 1, 2, 3 have the property that each divides the sum of...
 5.5.53: Show that the following statement is false.If A and B are two sets ...
 5.5.54: (a) Prove that if a 2 and n 1 are integers such that a2 + 1 = 2n, t...
 5.5.55: Prove that there do not exist positive integers a and n such that a...
 5.5.56: Let x, y R+. Use a proof by contradiction to prove that if x < y, t...
 5.5.57: The kings daughter had three suitors and couldnt decide which one t...
 5.5.58: Prove that if a, b, c, d are four real numbers, then at most four o...
 5.5.59: Evaluate the proposed proof of the following result.Result The numb...
 5.5.60: (a) Let n be a positive integer. Show that every integer m with 1 m...
 5.5.61: Prove that the sum of the irrational numbers 2, 3 and 5 is also irr...
 5.5.62: Let a1, a2,..., ar be odd integers where ai > 1 for i = 1, 2,...,r....
 5.5.63: Below is given a proof of a result. What result is proved?Proof Let...
 5.5.64: Evaluate the proposed proof of the following result.Result If x is ...
 5.5.65: Prove that there exist four distinct real numbers a, b, c, d such t...
 5.5.66: Below is given a proof of a result. What result is proved?Proof Let...
Solutions for Chapter 5: Existence and Proof by Contradiction
Full solutions for Mathematical Proofs: A Transition to Advanced Mathematics  3rd Edition
ISBN: 9780321797094
Solutions for Chapter 5: Existence and Proof by Contradiction
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 5: Existence and Proof by Contradiction includes 66 full stepbystep solutions. This textbook survival guide was created for the textbook: Mathematical Proofs: A Transition to Advanced Mathematics, edition: 3. Mathematical Proofs: A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780321797094. Since 66 problems in chapter 5: Existence and Proof by Contradiction have been answered, more than 5613 students have viewed full stepbystep solutions from this chapter.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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).

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).

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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

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).

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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·

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.