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Solutions for Chapter 3.7: Proof by Contradiction

Discrete Mathematics | 1st Edition | ISBN: 9781577667308 | Authors: Gary Chartrand, Ping Zhang

Full solutions for Discrete Mathematics | 1st Edition

ISBN: 9781577667308

Discrete Mathematics | 1st Edition | ISBN: 9781577667308 | Authors: Gary Chartrand, Ping Zhang

Solutions for Chapter 3.7: Proof by Contradiction

Since 15 problems in chapter 3.7: Proof by Contradiction have been answered, more than 13671 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics, edition: 1. Discrete Mathematics was written by and is associated to the ISBN: 9781577667308. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 3.7: Proof by Contradiction includes 15 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
  • Associative Law (AB)C = A(BC).

    Parentheses can be removed to leave ABC.

  • Block matrix.

    A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

  • Cross product u xv in R3:

    Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

  • Echelon matrix U.

    The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • Factorization

    A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

  • Fibonacci numbers

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

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Full row rank r = m.

    Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

  • 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.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Left inverse A+.

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

  • Linear transformation T.

    Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

  • 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.

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • 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.

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