 4.4.1: Let a and b be integers, where a = 0. Prove that if a  b, then a2 ...
 4.4.2: Let a, b Z, where a = 0 and b = 0. Prove that if a  b and b  a, t...
 4.4.3: Let m Z.(a) Give a direct proof of the following: If 3  m, then 3 ...
 4.4.4: Let x, y Z. Prove that if 3 x and 3 y, then 3  (x 2 y2). 4
 4.4.5: Let a, b, c Z, where a = 0. Prove that if a bc, then a b and a c...
 4.4.6: Let a Z. Prove that if 3  2a, then 3  a.
 4.4.7: Let n Z. Prove that 3  (2n2 + 1) if and only if 3 n.
 4.4.8: In Result 4.4, it was proved for an integer x that if 2  (x 2 1), ...
 4.4.9: (a) Let x Z. Prove that if 2  (x 2 5), then 4  (x 2 5).(b) Give a...
 4.4.10: Let n Z. Prove that 2  (n4 3) if and only if 4  (n2 + 3).
 4.4.11: Prove that for every integer n 8, there exist nonnegative integers ...
 4.4.12: In Result 4.7, it was proved for integers x and y that 4  (x 2 y2)...
 4.4.13: Prove that if a, b, c Z and a2 + b2 = c2, then 3  ab.
 4.4.14: Let a, b, n Z, where n 2. Prove that if a b (mod n), then a2 b2 (mo...
 4.4.15: Let a, b, c, n Z, where n 2. Prove that if a b (mod n) and a c (mod...
 4.4.16: Let a, b Z. Prove that if a2 + 2b2 0 (mod 3), then either a and b a...
 4.4.17: (a) Prove that if a is an integer such that a 1 (mod 5), then a2 1 ...
 4.4.18: Let m, n N such that m 2 and m  n. Prove that if a and b are integ...
 4.4.19: Let a, b Z. Show that if a 5 (mod 6) and b 3 (mod 4), then 4a + 6b ...
 4.4.20: (a) Result 4.12 states: Let n Z. If n2 n (mod 3), then n 0 (mod 3) ...
 4.4.21: Let a Z. Prove that a3 a (mod 3).
 4.4.22: Let n Z. Prove each of the statements (a)(f).(a) If n 0 (mod 7), th...
 4.4.23: Prove for any set S = {a, a + 1,..., a + 5} of six integers where 6...
 4.4.24: Let x and y be even integers. Prove that x 2 y2 (mod 16) if and onl...
 4.4.25: Let x, y R. Prove that if x 2 4x = y2 4y and x = y, then x + y = 4.
 4.4.26: Let a, b and m be integers. Prove that if 2a + 3b 12m + 1, then a 3...
 4.4.27: Let x R. Prove that if 3x 4 + 1 x 7 + x 3, then x > 0.
 4.4.28: Prove that if r is a real number such that 0 < r < 1, then 1r(1r) 4.
 4.4.29: Prove that if r is a real number such that r 1 < 1, then 4r(4r) 1.
 4.4.30: Let x, y R. Prove that x y=xy.
 4.4.31: Prove for every two real numbers x and y that x + yxy.
 4.4.32: (a) Recall that r > 0 for every positive real number r. Prove that ...
 4.4.33: The geometric mean of three positive real numbers a, b and c is 3 a...
 4.4.34: Prove for every three real numbers x, y and z that x zx y+y z.
 4.4.35: Prove that if x is a real number such that x(x + 1) > 2, then x < 2...
 4.4.36: Prove for every positive real number x that 1 + 1x4 1x + 1x3 .
 4.4.37: Prove for x, y,z R that x 2 + y2 + z2 x y + x z + yz.
 4.4.38: Let a, b, x, y R and r R+. Prove that if x a < r/2 and y b < r/...
 4.4.39: Prove that if a, b, c, d R, then (ab + cd) 2 (a2 + c2)(b2 + d2).
 4.4.40: Let A and B be sets. Prove that A B = (A B) (B A) (A B).
 4.4.41: In Result 4.21, it was proved for sets A and B that A B = A if and ...
 4.4.42: Let A and B be sets. Prove that A B = A if and only if A B.
 4.4.43: (a) Give an example of three sets A, B and C such that A B = A C bu...
 4.4.44: Prove that if A and B are sets such that A B = , then A = or B = .
 4.4.45: Let A = {n Z : n 1 (mod 2)} and B = {n Z : n 3 (mod 4)}. Prove that...
 4.4.46: Let A and B be sets. Prove that A B = A B if and only if A = B.
 4.4.47: Let A = {n Z : n 2 (mod 3)} and B = {n Z : n 1 (mod 2)}.(a) Describ...
 4.4.48: Let A = {n Z : 2  n} and B = {n Z : 4  n}. Let n Z. Prove that n ...
 4.4.49: Prove for every two sets A and B that A = (A B) (A B).
 4.4.50: Prove for every two sets A and B that A B, B A and A B are pairwise...
 4.4.51: Let A and B be subsets of a universal set. Which of the following i...
 4.4.52: Prove that A B = B A for every two sets A and B (Theorem 4.22(1b)).
 4.4.53: Prove that A (B C) = (A B) (A C) for every three sets A, B and C (T...
 4.4.54: Prove that A B = A B for every two sets A and B (Theorem 4.22(4b)).
 4.4.55: Let A, B and C be sets. Prove that (A B) (A C) = A (B C).
 4.4.56: Let A, B and C be sets. Prove that (A B) (A C) = A (B C).
 4.4.57: Let A, B and C be sets. Use Theorem 4.22 to prove that A (B C) = (A...
 4.4.58: Let A, B and C be sets. Prove that A (B C) = (A B) (A C).
 4.4.59: Show for every three sets A, B and C that A (B C) = (A C) (A  B).
 4.4.60: For A = {x, y}, determine A P(A)
 4.4.61: For A = {1} and B = {2}, determine P(A B) and P(A) P(B).
 4.4.62: Let A and B be sets. Prove that A B = if and only if A = or B = .
 4.4.63: For sets A and B, find a necessary and sufficient condition for A B...
 4.4.64: For sets A and B, find a necessary and sufficient condition for (A ...
 4.4.65: Let A, B and C be nonempty sets. Prove that A C B C if and only if ...
 4.4.66: Result 4.23 states that if A, B,C and D are sets such that A C and ...
 4.4.67: Let A, B and C be sets. Prove that A (B C) = (A B) (A C).
 4.4.68: Let A, B,C and D be sets. Prove that (A B) (C D) = (A C) (B D).
 4.4.69: Let A, B,C and D be sets. Prove that (A B) (C D) (A C) (B D).
 4.4.70: Let A and B be sets. Show, in general, that A B = A B.
 4.4.71: Let n Z. Prove that 5  n2 if and only if 5  n.
 4.4.72: Prove for integers a and b that 3  ab if and only if 3  a or 3  b.
 4.4.73: Prove that if n is an odd integer, then 8  [n2 + (n + 6)2 + 6].
 4.4.74: Prove that if n is an odd integer, then 8  [n2 + (n + 6)2 + 6].
 4.4.75: Prove that if n is an odd integer, then 8  (n4 + 4n2 + 11).
 4.4.76: Let n, m Z. Prove that if n 1 (mod 2) and m 3 (mod 4), then n2 + m ...
 4.4.77: Find two distinct positive integer values of a for which the follow...
 4.4.78: Prove for every two positive real numbers a and b that ab + ba 2.
 4.4.79: Prove the following: Let x R. If x(x 5) = 4, then 5x 2 4 = 1 implie...
 4.4.80: Let x, y R. Prove that if x < 0, then x 3 x 2 y x 2 y x y2.
 4.4.81: Prove that 3  (n3 4n) for every integer n.
 4.4.82: Evaluate the proposed proof of the following result.Result Let x, y...
 4.4.83: Below is given a proof of a result. What result is proved?Proof Ass...
 4.4.84: A proof of the following result is given.Result Let n Z. If n4 is e...
 4.4.85: Given below is an attempted proof of a result.Proof First, we show ...
 4.4.86: Evaluate the proposed proof of the following result.Result Let x, y...
 4.4.87: Evaluate the proposed proof of the following result.Result Let x, y...
 4.4.88: Evaluate the proposed proof of the following result.Result For ever...
 4.4.89: Prove that for every three integers a, b and c, the sum a b+a c...
 4.4.90: Prove for every four real numbers a, b, c and d that ac + bd a2 + b...
 4.4.91: Prove that for every real number x, sin6 x + 3 sin2 x cos2 x + cos6...
 4.4.92: Let a Z. Prove that if 6  a and 10  a, then 15  a.
 4.4.93: Let A = {x}. Give an example of a set concerning the set A to which...
 4.4.94: Let a, b Z. Prove that if a b (mod 2) and b a (mod 3), then a b (mo...
 4.4.95: Let a, b, c R. Prove that 3 2 (a2 + b2 + c2 + 1) a(b + 1) + b(c + 1...
 4.4.96: Prove that if a, b and c are positive real numbers, then (a + b + c...
 4.4.97: Let T = {1, 2,..., 8}.(a) Determine the elements of the set A = {a ...
 4.4.98: Consider the open sentenceP(m):5m + 1 = a2 for some a Z,where m N. ...
 4.4.99: Let a1, a2,..., an (n 3) be n integers such that ai+1 ai 1 for 1 ...
Solutions for Chapter 4: More on Direct Proof and Proof by Contrapositive
Full solutions for Mathematical Proofs: A Transition to Advanced Mathematics  3rd Edition
ISBN: 9780321797094
Solutions for Chapter 4: More on Direct Proof and Proof by Contrapositive
Get Full SolutionsMathematical Proofs: A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780321797094. Chapter 4: More on Direct Proof and Proof by Contrapositive includes 99 full stepbystep solutions. This textbook survival guide was created for the textbook: Mathematical Proofs: A Transition to Advanced Mathematics, edition: 3. Since 99 problems in chapter 4: More on Direct Proof and Proof by Contrapositive have been answered, more than 6107 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of 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).

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = 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.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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 .

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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·

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