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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 74EQ

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Let R be the relation R = {(a, c). (a, d), (b, a), (c, a). (c, b), (d, b)).
What is the matrix representation of this relation, given that the labels on the matrix rows and on the matrix columns are in alphabetical
order?
4
24"
"DIAGONAL
CLASS
O
O
O
/0 0 1 1
1000
1 1 0 0
0001/
O
0 0 0 1
1000
1 100
0001,
/0 0 0
1000
1 1 0
0
0 1 0 1
1
0 0 1 1
1000
11 00
0 1 0 0
NEXT >
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Let A = {1, 2, 3, 4, 6, 12, 24} and R be a relation on A such that xRy if and only if xdivides y.Find the matrix of the relation R.(i) Show that R is a partial order on A.Draw the digraph of R follow by its Hasse diagram.Determine least upper bound of {2, 3}. Determine the greatest lower bound of {4, 6}.
Let R be a relation on the set A = {0, 1, 4, 8} defined by xRy if and only if x + y < 8.(i) List the ordered pairs which belong to the relation R, Draw the digraph of R, Represent R in matrix form and Determine the in-degree and out-degree of each vertex.
1. Create (a) the ordered pairs, (b) graphical representation, and (c) the relation matrix, for the Relation R ‘is a factor of’, on set B{1,2,3,6}. (d) Classify the relation R as reflexive, irreflexive, symmetric, antisymmetric, or transitive.
a ) Let R be the relation on the set A={1,2,3,4} defined by aRb if and only if 2a>b+1. Find the matrix representing R∘R. b) Suppose that the relation R is defined on the set Z where aRb means a= ±b. Show that R is an equivalence relation.i would like to get a non handwriting answer to be easy to cope please
Given the following relation, { ( A, A ), ( A, B ), ( A, E ), ( B, B ), ( B, E ), ( C, A ), ( C, B ), ( C, C ), ( C, D ), ( D, A ), ( D, B ), ( D, D ), ( E, C ), ( E, D ), ( E, E ) }i) Draw the digraph of the relation, ii) construct the matrix diagram for the relation, and iii) why or why not is the relation reflexive, symmetric, antisymmetric, transitive?
Given that T is a relation on the set {1, 2, 3, 4}, and the set of ordered pairs is: T = {(1, 1), (1, 2), (1, 4), (2, 1), (2, 3), (3, 2), (3, 3), (3, 4), (4, 1), (4, 3), (4, 4)}. Can you draw T as a directed graph? Can you show T as a matrix with elements that are zeros and ones? Is T reflexive? Can you explain why?
Let A = {0,1,2,3} and R a relation over A: R = {(0,0),(0,1),(0,3),(1,1),(1,0),(2,3),(3,3)} Check whether R is an equivalence relation or a partial order. Give a counterexample in each case in which the relation does not satisfy one of the properties of being an equivalence relation or a partial order.
Let the set A = {2, 3, 8, 12} and the relation R defined as the following R = { (2, 2), (2, 8), (2, 12), (3, 3), (3, 12), (8, 8), (12, 12)} A. Represent R with a matrix (considering the elements of the set A listed in the same order as above) B. Determine the properties of the relation R (reflexive, symmetric, antisymetric, transitive, and/ or equivalence). C. Which of the followings describes R? 1. R = {(a, b) | a divides b} 2. R = {(a, b) | a > b} 3. R = {(a, b) | a + b < 15
How many relations are there on the set {a, b}. Show all possible relations on the set {a, b} in matrix format.
Consider the relation R defined on the set X={a,b,c,d} and Y={1,2,3,4} from X to Y, where R={(?, 1); (?, 3); (?, 2); (?, 3); (?, 4); (?, 2)}.(i)Deduce the matrix of the complementary relation, ???, clearly outline and comment on your result.(ii) A relation T is defined on the set Y above from Y to Y as ? ={(1,1); (4,2); (1,3); (2,4); (2,1); (3,2); (3,3); (3,4); (1,2); (2,3); (4,4); (4,1)}, compute and analyze ????−1. (b) Given the functions f and g de defined by ?(?) =6?+57?−5, ? ≠57; ??? ?(?) = 2?2 − 3? + 4. Critically analyze and deduce the formula defining the (i) composition function gof (ii) inverse function ?−1(c)Apply the Runge-Kutta formula as a tool for solving and analyzing numerical differential equation, critically compute and analyze the numerical solution of ?′ = ?(?, ?) = 1 + ?2, y(0) = 0. Computing for the first step, with ℎ = 0.5
Let the set A = {1, 3, 5, 9} and the relation R defined as the following R = {(1, 1), (1, 3), (1, 5), (1, 9), (3, 3), (3, 9), (5, 5), (9, 9)} A. Represent R with a matrix (considering the elements of the set A listed in the same order as above) B. Determine with justification the properties of the relation R (reflexive, symmetric, antisymmetric, transitive, and/ or equivalence) C. Which of the followings describes R? I. R = {(a, b) | a, b A and a < b} II. R = {(a, b) | a, b A and a divides b} III. R = {(a, b) | a, b A and a + b < 20}

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