 7.4.1: When asked what it means to say that set A has the same cardinality...
 7.4.2: Show that there are as many squares as there are numbers by exhibit...
 7.4.3: Show that there are as many squares as there are numbers by exhibit...
 7.4.4: Let O be the set of all odd integers. Prove that O has the same car...
 7.4.5: Let 25Z be the set of all integers that are multiples of 25. Prove ...
 7.4.6: Use the functions I and J defined in the paragraph following Exampl...
 7.4.7: a. Check that the formula for F given at the end of Example 7.4.2 p...
 7.4.8: Use the result of exercise 3 to prove that 3Z is countable.
 7.4.9: Show that the set of all nonnegative integers is countable by exhib...
 7.4.10: In 1014, S denotes the set of real numbers strictly between 0 and 1...
 7.4.11: In 1014, S denotes the set of real numbers strictly between 0 and 1...
 7.4.12: In 1014, S denotes the set of real numbers strictly between 0 and 1...
 7.4.13: In 1014, S denotes the set of real numbers strictly between 0 and 1...
 7.4.14: In 1014, S denotes the set of real numbers strictly between 0 and 1...
 7.4.15: Show that the set of all bit strings (strings of 0s and 1s) is coun...
 7.4.16: Show that Q, the set of all rational numbers, is countable
 7.4.17: Show that the set Q of all rational numbers is dense along the numb...
 7.4.18: Must the average of two irrational numbers always be irrational? Pr...
 7.4.19: Show that the set of all irrational numbers is dense along the numb...
 7.4.20: Give two examples of functions from Z to Z that are onetoone but n...
 7.4.21: Give two examples of functions from Z to Z that are onto but not on...
 7.4.22: Define a function g: Z+ Z+ Z+ by the formula g(m, n) = 2m3n for all...
 7.4.23: a. Explain how to use the following diagram to show that Znonneg Zn...
 7.4.24: Prove that the function H defined analytically in exercise 23b is a...
 7.4.25: Prove that 0.1999 ... = 0.2
 7.4.26: Prove that any infinite set contains a countably infinite subset.
 7.4.27: If A is any countably infinite set, B is any set, and g: A B is ont...
 7.4.28: Prove that a disjoint union of any finite set and any countably inf...
 7.4.29: Prove that a union of any two countably infinite sets is countably ...
 7.4.30: Use the result of exercise 29 to prove that the set of all irration...
 7.4.31: Use the results of exercises 28 and 29 to prove that a union of any...
 7.4.32: Prove that Z Z, the Cartesian product of the set of integers with i...
 7.4.33: Use the results of exercises 27, 31, and 32 to prove the following:...
 7.4.34: Let P(S) be the set of all subsets of set S, and let T be the set o...
 7.4.35: Let S be a set and let P(S) be the set of all subsets of S. Show th...
 7.4.36: The SchroederBernstein theorem states the following: If A and B are...
 7.4.37: Prove that if A and B are any countably infinite sets, then A B is ...
 7.4.38: Suppose A1, A2, A3,... is an infinite sequence of countable sets. R...
Solutions for Chapter 7.4: Cardinality with Applications to Computability
Full solutions for Discrete Mathematics with Applications  4th Edition
ISBN: 9780495391326
Solutions for Chapter 7.4: Cardinality with Applications to Computability
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 38 problems in chapter 7.4: Cardinality with Applications to Computability have been answered, more than 51826 students have viewed full stepbystep solutions from this chapter. Chapter 7.4: Cardinality with Applications to Computability includes 38 full stepbystep solutions. This textbook survival guide was created for the textbook: Discrete Mathematics with Applications , edition: 4. Discrete Mathematics with Applications was written by and is associated to the ISBN: 9780495391326.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

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

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

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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.

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.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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.

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