- 10.1.1: The ________ of A and B consists of all elements in either A or B o...
- 10.1.2: The ________ of A with B consists of all elements in both A and B.
- 10.1.3: True or False The intersection of two sets is always a subset of th...
- 10.1.4: True or False If A is a set, the complement of A is the set of all ...
- 10.1.5: If each element of a set A is also an element of a set B, we say th...
- 10.1.6: If the number of elements in a set is a nonnegative integer, we say...
- 10.1.7: If A and B are finite sets, the Counting Formula states that n(A B)...
- 10.1.8: True or False If a task consists of a sequence of three choices in ...
- 10.1.9: Write down all the subsets of 5a, b, c, d6.
- 10.1.10: Write down all the subsets of 5a, b, c, d, e6.
- 10.1.11: If and find n1A B2
- 10.1.12: If and find n1A B2.
- 10.1.13: If and find n1A2.
- 10.1.14: If and find n1A2.
- 10.1.15: How many are in set A?
- 10.1.16: How many are in set B?
- 10.1.17: How many are in A or B?
- 10.1.18: How many are in A and B?
- 10.1.19: How many are in A but not C?
- 10.1.20: How many are not in A?
- 10.1.21: How many are in A and B and C?
- 10.1.22: How many are in A or B or C?
- 10.1.23: Shirts and Ties A man has 5 shirts and 3 ties. How many different s...
- 10.1.24: Blouses and Skirts A woman has 5 blouses and 8 skirts. How many dif...
- 10.1.25: Four-digit Numbers How many four-digit numbers can be formed using ...
- 10.1.26: Five-digit Numbers How many five-digit numbers can be formed using ...
- 10.1.27: Analyzing Survey Data In a consumer survey of 500 people, 200 indic...
- 10.1.28: Analyzing Survey Data In a student survey, 200 indicated that they ...
- 10.1.29: Analyzing Survey Data In a survey of 100 investors in the stock mar...
- 10.1.30: Classifying Blood Types Human blood is classified as either Blood i...
- 10.1.31: Demographics The following data represent the marital status of mal...
- 10.1.32: Demographics The following data represent the marital status of fem...
- 10.1.33: Stock Portfolios As a financial planner, you are asked to select on...
- 10.1.34: Make up a problem different from any found in the text that require...
- 10.1.35: Investigate the notion of counting as it relates to infinite sets. ...
Solutions for Chapter 10.1: Counting
Full solutions for College Algebra | 9th Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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.
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.
Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.
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.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.