 7.5.7.5.1: How many elements are in A I U A2 if there are 12 elements in A I ,...
 7.5.7.5.2: There are 345 students at a college who have taken a course in calc...
 7.5.7.5.3: A survey of households in the United States reveals that 96% have a...
 7.5.7.5.4: A marketing report concerning personal computers states that 650,00...
 7.5.7.5.5: Find the number of elements in A I U A2 U A3 ifthere are 100 elemen...
 7.5.7.5.6: Find the number of elements in A I U A2 U A3 if there are 100 eleme...
 7.5.7.5.7: There are 2504 computer science students at a school. Of these, 187...
 7.5.7.5.8: In a survey of270 college students, it is found that 64 like brusse...
 7.5.7.5.9: How many students are enrolled in a course either in calculus, disc...
 7.5.7.5.10: Find the number of positive integers not exceeding 100 that are not...
 7.5.7.5.11: Find the number of positive integers not exceeding 100 that are eit...
 7.5.7.5.12: Find the number of positive integers not exceeding 1000 that are ei...
 7.5.7.5.13: How many bit strings of length eight do not contain six consecutive...
 7.5.7.5.14: How many permutations of the 26 letters of the English alphabet do ...
 7.5.7.5.15: How many permutations ofthe 10 digits either begin with the 3 digit...
 7.5.7.5.16: How many elements are in the union of four sets if each of the sets...
 7.5.7.5.17: How many elements are in the union of four sets if the sets have 50...
 7.5.7.5.18: How many terms are there in the formula for the number of elements ...
 7.5.7.5.19: Write out the explicit formula given by the principle of inclusion...
 7.5.7.5.20: How many elements are in the union of five sets if the sets contain...
 7.5.7.5.21: Write out the explicit formula given by the principle of inclusion...
 7.5.7.5.22: Prove the principle of inclusionexclusion using mathematical induc...
 7.5.7.5.23: Let E 1, E2, and E3 be three events from a sample space S. Find a f...
 7.5.7.5.24: Find the probability that when a fair coin is flipped five times ta...
 7.5.7.5.25: Find the probability that when four numbers from 1 to 100, inclusiv...
 7.5.7.5.26: Find a formula for the probability of the union of four events in a...
 7.5.7.5.27: Find a formula for the probability of the union of five events in a...
 7.5.7.5.28: Find a formula for the probability of the union of n events in a sa...
 7.5.7.5.29: Find a formula for the probability of the union ofn events in a sam...
Solutions for Chapter 7.5: Advanced Counting Techniques
Full solutions for Discrete Mathematics and Its Applications  6th Edition
ISBN: 9780073229720
Solutions for Chapter 7.5: Advanced Counting Techniques
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Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

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].

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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.

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

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).