Mathematics: A Discrete Introduction
3rd Edition
ISBN: 9780840049421
Author: Edward A. Scheinerman
Publisher: Cengage Learning
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Chapter 1.5, Problem 5.16E
Let a and b be integers. Prove that
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Find a basis and dimension for the null space of the following matrix:
3 -2 0 7
-2
1-1 1
5 3 19-2
8
06
1
-2 -4
-5-6
-9
4-6
11
6
Find a basis and dimension for the column space of the same matrix (above).
3. (i) Below is the R code for performing a X2 test on a 2×3 matrix of categorical
variables called TestMatrix:
chisq.test(Test Matrix)
(a) Assuming we have a significant result for this procedure, provide the R
code (including any required packages) for an appropriate post hoc test.
(b) If we were to apply this technique to a 2 × 2 case, how would we adapt
the code in order to perform the correct test?
(ii) What procedure can we use if we want to test for association when we
have ordinal variables? What code do we use in R to do this? What package
does this command belong to?
(iii) The following code contains the initial steps for a scenario where we are
looking to investigate the relationship between age and whether someone owns
a car by using frequencies. There are two issues with the code - please state
these.
Row3<-c(75,15)
Row4<-c(50,-10)
MortgageMatrix<-matrix(c(Row1, Row4), byrow=T, nrow=2,
MortgageMatrix
dimnames=list(c("Yes", "No"), c("40 or older","<40")))…
Chapter 1 Solutions
Mathematics: A Discrete Introduction
Ch. 1.1 - Simplify the following algebraic expression:...Ch. 1.2 - Prob. 2.1ECh. 1.3 - Prob. 3.1ECh. 1.3 - Prob. 3.2ECh. 1.3 - Prob. 3.3ECh. 1.3 - Prob. 3.4ECh. 1.3 - Prob. 3.5ECh. 1.3 - Prob. 3.6ECh. 1.3 - Prob. 3.7ECh. 1.3 - Prob. 3.8E
Ch. 1.3 - Prob. 3.9ECh. 1.3 - Prob. 3.10ECh. 1.3 - Prob. 3.11ECh. 1.3 - Prob. 3.12ECh. 1.3 - Prob. 3.13ECh. 1.3 - Prob. 3.14ECh. 1.4 - Prob. 4.1ECh. 1.4 - Prob. 4.2ECh. 1.4 - Prob. 4.3ECh. 1.4 - Prob. 4.4ECh. 1.4 - Prob. 4.5ECh. 1.4 - Prob. 4.6ECh. 1.4 - Prob. 4.7ECh. 1.4 - Prob. 4.8ECh. 1.4 - Prob. 4.9ECh. 1.4 - Prob. 4.10ECh. 1.4 - Prob. 4.11ECh. 1.4 - Prob. 4.12ECh. 1.5 - Prove that the sum of two odd integers is even.Ch. 1.5 - Prove that the sum of an odd integer and an even...Ch. 1.5 - Prove that if n is an odd integer, then n is also...Ch. 1.5 - Prove that the product of two even integers is...Ch. 1.5 - Prove that the product of an even integer and an...Ch. 1.5 - Prove that the product of two odd integers is odd.Ch. 1.5 - Prove that the square of an odd integer is odd.Ch. 1.5 - Prove that the cube of an odd integer is odd.Ch. 1.5 - Suppose a, b, and c are integers. Prove that if ab...Ch. 1.5 - Suppose a, b, and c are integers. Prove that if...Ch. 1.5 - Suppose a, b, d, x, and y are integers. Prove that...Ch. 1.5 - Suppose a, b, c, and d are integers. Prove that if...Ch. 1.5 - Let x be an integer. Prove that x is odd if and...Ch. 1.5 - Let x be an integers. Prove that x is odd if and...Ch. 1.5 - Let x be an integer. Prove that 0x if and only if...Ch. 1.5 - Let a and b be integers. Prove that ab if and only...Ch. 1.5 - Let a be a number with a1. Prove that a number x...Ch. 1.5 - Prove that the difference between consecutive...Ch. 1.5 - Let a be a perfect square. Prove that a is the...Ch. 1.5 - For real numbers a and b, prove that if 0ab, then...Ch. 1.5 - Prove that the difference between distinct,...Ch. 1.5 - Prove that an integer is odd if and only if it is...Ch. 1.5 - Suppose you are asked to prove a statement of the...Ch. 1.5 - Suppose you are asked to prove a statement of the...Ch. 1.6 - Disprove: If a and b are integers with ab, then...Ch. 1.6 - Disprove: If a and b are nonnegative integers with...Ch. 1.6 - Disprove: If a, b, and c are positive integers...Ch. 1.6 - Disprove: If a, b, and c are positive integers,...Ch. 1.6 - Disprove: If p and q are prime, then p+q is...Ch. 1.6 - Disprove: If p is prime, then 2p1 is also prime.Ch. 1.6 - Prob. 6.7ECh. 1.6 - An integer is a palindrome if it reads the same...Ch. 1.6 - Prob. 6.9ECh. 1.6 - Prob. 6.10ECh. 1.6 - Prob. 6.11ECh. 1.6 - Prob. 6.12ECh. 1.6 - Prob. 6.13ECh. 1.7 - Prob. 7.1ECh. 1.7 - Prob. 7.2ECh. 1.7 - Prob. 7.3ECh. 1.7 - Prob. 7.4ECh. 1.7 - Prob. 7.5ECh. 1.7 - Prob. 7.6ECh. 1.7 - Prob. 7.7ECh. 1.7 - Prob. 7.8ECh. 1.7 - Prob. 7.9ECh. 1.7 - Prob. 7.10ECh. 1.7 - Prob. 7.11ECh. 1.7 - Prob. 7.12ECh. 1.7 - Prob. 7.13ECh. 1.7 - Prob. 7.14ECh. 1.7 - Prob. 7.15ECh. 1.7 - Prob. 7.16ECh. 1.7 - Prob. 7.17ECh. 1.7 - Prob. 7.18ECh. 1.7 - Prove that xy can be reexpressed in terms of just ...Ch. 1.7 - Prob. 7.20ECh. 1 - Prob. 1STCh. 1 - Prob. 2STCh. 1 - Prob. 3STCh. 1 - Prob. 4STCh. 1 - Prob. 5STCh. 1 - Prob. 6STCh. 1 - Prob. 7STCh. 1 - Prob. 8STCh. 1 - Prob. 9STCh. 1 - Prob. 10STCh. 1 - Prob. 11STCh. 1 - Prob. 12STCh. 1 - Prob. 13STCh. 1 - Prob. 14STCh. 1 - Prob. 15STCh. 1 - Prob. 16STCh. 1 - Prob. 17STCh. 1 - Prob. 18STCh. 1 - Prob. 19STCh. 1 - Prob. 20ST
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- Describe the situation in which Fisher’s exact test would be used?(ii) When do we use Yates’ continuity correction (with respect to contingencytables)?[2 Marks] 2. Investigate, checking the relevant assumptions, whether there is an associationbetween age group and home ownership based on the sample dataset for atown below:Home Owner: Yes NoUnder 40 39 12140 and over 181 59Calculate and evaluate the effect size.arrow_forwardsolve these pleasearrow_forwardA factorization A = PDP 1 is not unique. For A= 7 2 -4 1 1 1 5 0 2 1 one factorization is P = D= and P-1 30 = Use this information with D₁ = to find a matrix P₁ such that - -1 -2 0 3 1 - - 1 05 A-P,D,P P1 (Type an integer or simplified fraction for each matrix element.)arrow_forward
- Matrix A is factored in the form PDP 1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 30 -1 - 1 0 -1 400 0 0 1 A= 3 4 3 0 1 3 040 3 1 3 0 0 4 1 0 0 003 -1 0 -1 Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) A basis for the corresponding eigenspace is { A. There is one distinct eigenvalue, λ = B. In ascending order, the two distinct eigenvalues are λ₁ ... = and 2 = Bases for the corresponding eigenspaces are { and ( ), respectively. C. In ascending order, the three distinct eigenvalues are λ₁ = = 12/2 = and 3 = Bases for the corresponding eigenspaces are {}, }, and { respectively.arrow_forwardN Page 0.6. 0.4. 0.2- -0.2- -0.4- -6.6 -5 W 10arrow_forwardDiagonalize the following matrix, if possible. 8 0 6 - 8 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. 8 0 OA. For P= D= 0 3 6 0 B. For P = D= 0 -6 8 0 C. For P = D= 0 - 8 D. The matrix cannot be diagonalized.arrow_forward
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