 3.4.1: Prove that if n is an integer, then n2 3n + 5 is odd.
 3.4.2: Prove that if n is an integer, then n3 n is even.
 3.4.3: Let n Z. Prove that if n2 + n = 0, then 2n+3n 12n is even.
 3.4.4: Prove that if n is an integer, then 3n + 1 and 5n + 2 are of opposi...
 3.4.5: (a) Let m and n be two integers. Prove that m+ n is even if and onl...
 3.4.6: Let m and n be two integers. Prove that 3mn is even if and only if ...
 3.4.7: Let m and n be two integers. Prove that 7m+3n is odd if and only if...
 3.4.8: Let m and n be two integers. Prove that if m and n are of opposite ...
 3.4.9: Let m and n be two integers. Prove that mn2 is odd if and only if m...
 3.4.10: Let m and n be two integers. Prove that mn and m+ n are both even i...
 3.4.11: Give a proof of Let n Z. If 2n 1 5, then n 3 and n 2. using (a) a...
 3.4.12: In the proof of Result 3.28 it was proved, for integers m and n, th...
 3.4.13: Give a proof of Result 3.29 where without loss of generality is use...
 3.4.14: Use a proof by cases (as in the proof of Result 3.29) to prove the ...
 3.4.15: Prove the following: Let A and B be sets. Then (A B) (B A) = (A B) ...
 3.4.16: Let S = {0, 1, 3, 4}. Prove that if x S, then x2 4x + 3 = 0 or x2 4...
Solutions for Chapter 3.4: Proof by Cases
Full solutions for Discrete Mathematics  1st Edition
ISBN: 9781577667308
Solutions for Chapter 3.4: Proof by Cases
Get Full SolutionsDiscrete Mathematics was written by and is associated to the ISBN: 9781577667308. Chapter 3.4: Proof by Cases includes 16 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 16 problems in chapter 3.4: Proof by Cases have been answered, more than 13779 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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).

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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.

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