- 19.19.1: There are four large groups of people, each with 1000 members. Any ...
- 19.19.2: Of the integers between 1 and 100 (inclusive) how many are divisibl...
- 19.19.3: Of the integers between 1 and 1,000,000 (inclusive) how many are no...
- 19.19.4: Let A, B, and C be finite sets. Prove or disprove: If jA [ B [ Cj D...
- 19.19.5: How many five-letter words can you make in which no two consecutive...
- 19.19.6: This problem asks you to give two proofs for 9 n D Xn kD0 .1/k n k ...
- 19.19.7: How many six-digit numbers do not have three consecutive digits the...
- 19.19.8: How many lattice paths are there through the grid in the figure tha...
- 19.19.9: Note the following: jA \ Bj D jAj C jBj jA [ Bj. Find a general for...
- 19.19.10: . Let A1; A2; : : : ; An be finite sets and let A D A1 [ A2 [ [ An....
- 19.19.11: This problem refines part (a) of the previous exercise. Again, let ...
- 19.19.12: This exercise is for those who have studied calculus. In this secti...
Solutions for Chapter 19: Inclusion-Exclusion
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
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).
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 n-l - An).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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.
Gram-Schmidt 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.
lA-II = 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 nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Solvable system Ax = b.
The right side b is in the column space of A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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.