Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
8th Edition
ISBN: 9781259676512
Author: Kenneth H Rosen
Publisher: McGraw-Hill Education
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Textbook Question
Chapter 9.2, Problem 16E
Display the table produced by applying the projection P1,2,3to Table 8.
| TABLE 8Flights. | |||||||||
| Airline | Flight_number | Gate | Detination | Departure_time | |||||
| Nadir | |||||||||
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Chapter 9 Solutions
Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
Ch. 9.1 - t the ordered pairs in the...Ch. 9.1 - a) List all the ordered pairs in the relation R =...Ch. 9.1 - each of these relations on the set {1, 2, 3, 4},...Ch. 9.1 - ermine whether the relationRon the set of all...Ch. 9.1 - ermine whether the relationRon the set of all Web...Ch. 9.1 - ermine whether the relationRon the set of all real...Ch. 9.1 - ermine whether the relationRon the set of all...Ch. 9.1 - w that the relationR=Oon a nonempty set S is...Ch. 9.1 - Show that the relationR=on the empty setS=is...Ch. 9.1 - e an example of a relation on a set that is a)...
Ch. 9.1 - Which relations in Exercise 3 are irreflexive?Ch. 9.1 - Which relations in Exercise 4 are irreflexive?Ch. 9.1 - Which relations in Exercise 5 are irreflexive?Ch. 9.1 - Which relations in Exercise 6 are irreflexive?Ch. 9.1 - Can a relation on a set be neither reflexive nor...Ch. 9.1 - Use quantifiers to express what it means for a...Ch. 9.1 - Give an example of an irreflexive relation on the...Ch. 9.1 - Which relations in Exercise 3 are asymmetric?Ch. 9.1 - Which relations in Exercise 4 are asymmetric?Ch. 9.1 - Which relations in Exercise 5 are asymmetric?Ch. 9.1 - Which relations in Exercise 6 are asymmetric?Ch. 9.1 - Must an asymmetric relation also be antisymmetric?...Ch. 9.1 - Use quantifiers to express what it means for...Ch. 9.1 - Give an example of an asymmetric relation on the...Ch. 9.1 - many different relations are there from a set...Ch. 9.1 - Rbe the relationR={(a,b)ab}on the set of integers....Ch. 9.1 - Rbe the relationR={(a,b) |adividesb} on the set of...Ch. 9.1 - Let R be the relation on the set of all states in...Ch. 9.1 - pose that the functionffromAtoBis a one-to-one...Ch. 9.1 - R1= {(1, 2), (2, 3), (3, 4)} andR2= {(1, 1), (1,...Ch. 9.1 - Abe the set of students at your school andBthe set...Ch. 9.1 - Rbe the relation {(1, 2), (1, 3), (2, 3), (2,4),...Ch. 9.1 - 33.LetRbe the relation on the set of people...Ch. 9.1 - rcises 34-38 deal with these relations on the set...Ch. 9.1 - rcises 34-38 deal with these relations on the set...Ch. 9.1 - rcises 34-38 deal with these relations on the set...Ch. 9.1 - rcises 34-38 deal with these relations on the set...Ch. 9.1 - rcises 34-38 deal with these relations on the set...Ch. 9.1 - d the relationsS2fori= 1, 2, 3,4, , 6i’here...Ch. 9.1 - Rbe the parent relation on the set of all people...Ch. 9.1 - Rbe the relation on the set of people with...Ch. 9.1 - R1andR2be the divides” and ‘is a multiple of...Ch. 9.1 - R1andR2be the “congruent modulo 3” and the...Ch. 9.1 - List the 16 different relations on the set {0,1}.Ch. 9.1 - How many of the 16 different relations on {0,1}...Ch. 9.1 - ch of the 16 relations on {o, 1}, which you listed...Ch. 9.1 - a) How many relations are there on the set...Ch. 9.1 - S be a set withnelements and letaandbbe distinct...Ch. 9.1 - How many relations are there on a set...Ch. 9.1 - How many transitive relations are there on a set...Ch. 9.1 - d the error in the “proof” of the following...Ch. 9.1 - pose thatRandSare reflexive relations on a setA....Ch. 9.1 - w that the relationRon a setAis symmetric if and...Ch. 9.1 - w that the relationRon a setAis antisymmetric if...Ch. 9.1 - w that the relationRon a setAis reflexive if and...Ch. 9.1 - w that the relationRon a setAis reflexive if and...Ch. 9.1 - Rbe a relation that is reflexive and transitive....Ch. 9.1 - Rbe the relation on the set {1, 2, 3,4 , 5}...Ch. 9.1 - Rbe a reflexive relation on a setA. Show thatRnis...Ch. 9.1 - Prob. 60ECh. 9.1 - Suppose that the relationRis irreflexive....Ch. 9.1 - ive a big-O estimate for the number of integer...Ch. 9.2 - List the triples in the relation {(a, b, c)|a,...Ch. 9.2 - ch 4-tuples are in the relation {(a,b, c, d)| a,...Ch. 9.2 - Prob. 3ECh. 9.2 - uming that no newn-tuples are added, find all the...Ch. 9.2 - Prob. 5ECh. 9.2 - uming that no new n-tuples are added, find a...Ch. 9.2 - Prob. 7ECh. 9.2 - Prob. 8ECh. 9.2 - 5-tuples in a 5-ary relation represent these...Ch. 9.2 - What do you obtain when you apply the selection...Ch. 9.2 - What do you obtain when you apply the selection...Ch. 9.2 - What do you obtain when you apply the selection...Ch. 9.2 - t do you obtain when you apply the selection...Ch. 9.2 - t do you obtain when you apply the...Ch. 9.2 - Prob. 15ECh. 9.2 - Display the table produced by applying the...Ch. 9.2 - play the table produced by applying the...Ch. 9.2 - many components are there in then-tuples in the...Ch. 9.2 - Construct the table obtained by applying the join...Ch. 9.2 - w that ifC1andC2are conditions that elements of...Ch. 9.2 - w that if C1andC2are conditions that elements...Ch. 9.2 - Prob. 22ECh. 9.2 - Prob. 23ECh. 9.2 - w that ifCis a condition that elements of the nary...Ch. 9.2 - w that ifRandSare bothn-ary relations,...Ch. 9.2 - Give an example to show that ifRandSare bothn-ary...Ch. 9.2 - e an example to show that ifRandSare bothn-ary...Ch. 9.2 - a) What are the operations that correspond to the...Ch. 9.2 - Prob. 29ECh. 9.2 - Prob. 30ECh. 9.2 - ermine whether there is a primary key for the...Ch. 9.2 - Show that ann-aryrelation with a primary key can...Ch. 9.2 - Prob. 33ECh. 9.2 - Prob. 34ECh. 9.2 - Prob. 35ECh. 9.2 - Prob. 36ECh. 9.2 - Prob. 37ECh. 9.2 - Prob. 38ECh. 9.2 - Prob. 39ECh. 9.2 - Show that if an item set is frequent in a set of...Ch. 9.2 - Prob. 41ECh. 9.3 - resent each of these relations on {1, 2, 3} with a...Ch. 9.3 - resent each of these relations on {1, 2,3, 4} with...Ch. 9.3 - List the ordered pairs in the relations on {1, 2,...Ch. 9.3 - t the ordered pairs in the relations on {1,2,3,4)...Ch. 9.3 - can the matrix representing a relationRon a setAbe...Ch. 9.3 - can the matrix representing a relationRon a setAbe...Ch. 9.3 - ermine whether the relations represented by the...Ch. 9.3 - Determine whether the relation represented by the...Ch. 9.3 - many nonzero entries does the matrix representing...Ch. 9.3 - many nonzero entries does the matrix representing...Ch. 9.3 - How can the matrixR, the complement of the...Ch. 9.3 - How can the matrix forR1, the inverse of the...Ch. 9.3 - LetRbe the relation represented by the matrix...Ch. 9.3 - R1andR2be relations on a setArepresented by the...Ch. 9.3 - Rbe the relation represented by the matrix...Ch. 9.3 - Rbe a relation on a set A withnelements. If there...Ch. 9.3 - Rbe a relation on a set A withnelements. If there...Ch. 9.3 - Draw the directed graphs representing each of the...Ch. 9.3 - Draw the directed graphs representing each of the...Ch. 9.3 - Draw the directed graph representing each of the...Ch. 9.3 - Draw the directed graph representing each of the...Ch. 9.3 - Draw the directed graph that represents the...Ch. 9.3 - Exercises 23-28 list the ordered pairs in the...Ch. 9.3 - Exercises 23-28 list the ordered pairs in the...Ch. 9.3 - Prob. 25ECh. 9.3 - Prob. 26ECh. 9.3 - Prob. 27ECh. 9.3 - Exercises 23-28 list the ordered pairs in the...Ch. 9.3 - can the directed graph of a relationRon a finite...Ch. 9.3 - How can the directed graph of a relationRon finite...Ch. 9.3 - ermine whether the relations represented by the...Ch. 9.3 - ermine whether the relations represented by the...Ch. 9.3 - LetRbe a relation on a setA, Explain how to use...Ch. 9.3 - Rbe a relation on a set A. Explain how to use the...Ch. 9.3 - w that ifMRis the matrix representing the...Ch. 9.3 - Prob. 36ECh. 9.4 - Rbe the relation on the set {o, 1, 2, 3}...Ch. 9.4 - LetRbe the relation{(a,b)ab}on the set of...Ch. 9.4 - Rbe the relation{(a,b)| adividesb} on the set of...Ch. 9.4 - How can the directed graph representing the...Ch. 9.4 - Exercises 5-7 draw the directed graph of the...Ch. 9.4 - Exercises 5-7 draw the directed graph of the...Ch. 9.4 - Prob. 7ECh. 9.4 - How can the directed graph representing the...Ch. 9.4 - d the directed graphs of the symmetric closures of...Ch. 9.4 - Find the smallest relation containing the relation...Ch. 9.4 - Prob. 11ECh. 9.4 - Suppose that the relationRon the finite setAis...Ch. 9.4 - Prob. 13ECh. 9.4 - Prob. 14ECh. 9.4 - n is it possible to define the ‘irreflexive...Ch. 9.4 - Prob. 16ECh. 9.4 - Prob. 17ECh. 9.4 - Prob. 18ECh. 9.4 - Rbe the relation on the set{1,2,3,4,5} containing...Ch. 9.4 - Rbe the relation that contains the pair (a,b)...Ch. 9.4 - Rbe the relation on the set of all students...Ch. 9.4 - Suppose that the relationRis reflexive. Show...Ch. 9.4 - Suppose that the relationRis symmetric. Show...Ch. 9.4 - pose that the relationRis irreflexive. Is the...Ch. 9.4 - Algorithm 1 to find the transitive closures of...Ch. 9.4 - Algorithm 1 to find the transitive closures of...Ch. 9.4 - Use Warshall’s algorithm to find the transitive...Ch. 9.4 - Warshall’s algorithm to find the transitive...Ch. 9.4 - d the smallest relation containing the relation...Ch. 9.4 - Finish the proof of the case whenabin Lemma 1.Ch. 9.4 - orithms have been devised that use Q(n2,8) bit...Ch. 9.4 - Devise an algorithm using the concept of interior...Ch. 9.4 - Adapt Algorithm 1 to find the reflexive closure of...Ch. 9.4 - pt Warshall’s algorithm to find the reflexive...Ch. 9.4 - Prob. 35ECh. 9.4 - Prob. 36ECh. 9.5 - Which of these relations on {0, 1, 2,3) are...Ch. 9.5 - ch of these relations on the set of all people are...Ch. 9.5 - ch of these relations on the set of all functions...Ch. 9.5 - ine three equivalence relations on the set of...Ch. 9.5 - Define three equivalence relations on the set of...Ch. 9.5 - ine three equivalence relations on the set of...Ch. 9.5 - Show that the relation of logical equivalence on...Ch. 9.5 - Rbe the relation on the set of all sets of real...Ch. 9.5 - pose thatAis a nonempty set, andfis a function...Ch. 9.5 - pose thatAis a nonempty set andRis an equivalence...Ch. 9.5 - w that the relationRconsisting of all pairs (x, y)...Ch. 9.5 - w that the relationRconsisting of all pairs(x,...Ch. 9.5 - w that the relationRconsisting of all pairs (x, y)...Ch. 9.5 - R be the relation consisting of all pairs (x,y)...Ch. 9.5 - Rbe the relation on the set of ordered pairs of...Ch. 9.5 - Let R be the relation on the set of ordered pairs...Ch. 9.5 - (Requires calculus) a) Show that the relationRon...Ch. 9.5 - Prob. 18ECh. 9.5 - Rbe the relation on the set of all URLs (or Web...Ch. 9.5 - Rbe the relation on the set of all people who have...Ch. 9.5 - Prob. 21ECh. 9.5 - Prob. 22ECh. 9.5 - Exercises 21-23 determine whether the relation...Ch. 9.5 - Determine whether the relations represented by...Ch. 9.5 - w that the relationRon the set of all bit stings...Ch. 9.5 - t are the equivalence classes of the equivalence...Ch. 9.5 - t are the equivalence classes of the equivalence...Ch. 9.5 - t are the equivalence classes of the equivalence...Ch. 9.5 - What is the equivalence class of the bit string...Ch. 9.5 - t are the equivalence classes of these bit strings...Ch. 9.5 - What are the equivalence classes of the bit...Ch. 9.5 - What are the equivalence classes of the bit...Ch. 9.5 - t are the equivalence classes of the bit strings...Ch. 9.5 - t are the equivalence classes of the bit strings...Ch. 9.5 - t is the congruence class [n]5(that is, the...Ch. 9.5 - What is the congruence class [4]mwhenmis a) 2? b)...Ch. 9.5 - Give a description of each of the congruence...Ch. 9.5 - t is the equivalence class of each of these...Ch. 9.5 - a) What is the equivalence class of(1,2)with...Ch. 9.5 - a) What is the equivalence class of (1, 2) with...Ch. 9.5 - ch of these collections of subsets are partitions...Ch. 9.5 - ch of these collections of subsets are partitions...Ch. 9.5 - ch of these collections of subsets are partitions...Ch. 9.5 - ch of these collections of subsets are partitions...Ch. 9.5 - Prob. 45ECh. 9.5 - ch of these are partitions of the set of real...Ch. 9.5 - t the ordered pairs in the equivalence relations...Ch. 9.5 - t the ordered pairs in the equivalence relations...Ch. 9.5 - w that the partition formed from congruence...Ch. 9.5 - w that the paron of the set of people living in...Ch. 9.5 - w that the partition of the set of bit strings of...Ch. 9.5 - Exercises 52 and 53,Rnrefers to the family of...Ch. 9.5 - Exercises 52 and 53,Rnrefers to the family of...Ch. 9.5 - pose thatR1andR2are equivalence relations on a...Ch. 9.5 - d the smallest equivalence relation on the set...Ch. 9.5 - pose thatR1andR2are equivalence relations on the...Ch. 9.5 - sider the equivalence relation fromExample...Ch. 9.5 - Each bead on a bracelet with three beads is either...Ch. 9.5 - Let R be the relation on the set of all colorings...Ch. 9.5 - a) LetRbe the relation on the set of functions...Ch. 9.5 - Determine the number of different equivalence...Ch. 9.5 - Determine the number of different equivalence...Ch. 9.5 - Do we necessarily get an equivalence relation when...Ch. 9.5 - Do we necessarily get an equivalence relation when...Ch. 9.5 - pose we useTheorem 2to form a partitionP froman...Ch. 9.5 - .Suppose we useTheorem 2to form an equivalence...Ch. 9.5 - ise an algorithm to find the smallest equivalence...Ch. 9.5 - p(n)denote the number of different equivalence...Ch. 9.5 - Use Exercise 68 to find the number of different...Ch. 9.6 - ch of these relations on {0,1,2,3) are partial...Ch. 9.6 - ch of these relations on {0,1,2,3} are partial...Ch. 9.6 - Prob. 3ECh. 9.6 - Prob. 4ECh. 9.6 - ch of these are posets? a)(Z,=) b)(Z,) c)(Z,)...Ch. 9.6 - Which of these are posets?a) (R, =)b) (R,<) c)...Ch. 9.6 - Determine whether the relations represented by...Ch. 9.6 - Determine whether the relations represented by...Ch. 9.6 - Exercises9-11determine whether the relation with...Ch. 9.6 - Exercises9-11determine whether the relation with...Ch. 9.6 - Exercises 9-11 determine whether the relation with...Ch. 9.6 - Prob. 12ECh. 9.6 - d the duals of these posets. a)({0,1,2},) b)(Z,)...Ch. 9.6 - ch of these pairs of elements are comparable in...Ch. 9.6 - Prob. 15ECh. 9.6 - Let S = {1,2,3,4). With respect to the...Ch. 9.6 - d the lexicographic ordering of thesen-tuples: a)...Ch. 9.6 - d the lexicographic ordering of these strings of...Ch. 9.6 - d the lexicographic ordering of the bit strings...Ch. 9.6 - w the Hasse diagram for the greater than or equal...Ch. 9.6 - w the Hasse Diagram for the less than or equal to...Ch. 9.6 - Prob. 22ECh. 9.6 - Prob. 23ECh. 9.6 - w the Hasse diagram for inclusion on the...Ch. 9.6 - Exercises 25-27 list all ordered pairs in the...Ch. 9.6 - Exercises 25-27 list all ordered pairs in the...Ch. 9.6 - Exercises 25-27 list all ordered pairs in the...Ch. 9.6 - What is the covering relation of the partial...Ch. 9.6 - What is the covering relation of the partial...Ch. 9.6 - What is the covering relation of the partial...Ch. 9.6 - w that a finite poset can be reconstructed from...Ch. 9.6 - wer these questions for the partial order...Ch. 9.6 - wer these questions for the poset ({3, 5,9, 15,...Ch. 9.6 - wer these questions for the poset ({2, 4, 6, 9,...Ch. 9.6 - wer these questions for the poset ({{1}, {2}, {4},...Ch. 9.6 - Prob. 36ECh. 9.6 - Show that lexicographic order is a partial...Ch. 9.6 - w that lexicographic order is a partial ordering...Ch. 9.6 - Suppose that (S,1) and (T,2) are posets. Show...Ch. 9.6 - a) Show that there is exactly one greatest element...Ch. 9.6 - a) Show that there is exactly one maximal element...Ch. 9.6 - a) Show that the least upper bound of a set in a...Ch. 9.6 - Determine whether the posets with these Hasse...Ch. 9.6 - Prob. 44ECh. 9.6 - Show that every nonempty finite subset of a...Ch. 9.6 - Show that if the poset (S,R) is a lattice then the...Ch. 9.6 - a company, the lattice model of information flow...Ch. 9.6 - Prob. 48ECh. 9.6 - Show that the set of all partitions of a set S...Ch. 9.6 - Show that every totally ordered set is a lattice.Ch. 9.6 - Show that every finite lattice has a least element...Ch. 9.6 - Give an example of an infinite lattice with a)...Ch. 9.6 - Prob. 53ECh. 9.6 - ermine whether each of these posets is...Ch. 9.6 - Prob. 55ECh. 9.6 - Show that dense poset with at least two elements...Ch. 9.6 - Show that the poset of rational numbers with the...Ch. 9.6 - Show that the set of strings of lowercase English...Ch. 9.6 - Prob. 59ECh. 9.6 - w that a finite nonempty poset has a maximal...Ch. 9.6 - Find a compatible total order for the poset with...Ch. 9.6 - d a compatible total order for the divisibility...Ch. 9.6 - Find all compatible total orderings for the poset...Ch. 9.6 - Find all compatible total orderings for the poset...Ch. 9.6 - Find all possible orders for completing the tasks...Ch. 9.6 - Schedule the tasks needed to build a house, by...Ch. 9.6 - Prob. 67ECh. 9 - Prob. 1RQCh. 9 - a) What is a reflexive relation? b) What is a...Ch. 9 - e an example of a relation on the set {1, 2,3,4}...Ch. 9 - a) How many reflexive relations are there on a set...Ch. 9 - a) Explain how ann-ary relation can be used to...Ch. 9 - a) Explain how to use a zero-one matrix to...Ch. 9 - a) Explain how to use a directed graph to...Ch. 9 - a) Define the reflexive closure and the symmetric...Ch. 9 - a) Define the transitive closure of a relation. b)...Ch. 9 - a) Define an equivalence relation. b) Which...Ch. 9 - a) Show that congruence modulo in is an...Ch. 9 - a) What are the equivalence classes of an...Ch. 9 - lain the relationship between equivalence...Ch. 9 - a) Define a partial ordering. b) Show that the...Ch. 9 - Explain how partial orderings on the...Ch. 9 - a) Explain how to construct the Hasse diagram of a...Ch. 9 - a) Define a maximal element of a poset and the...Ch. 9 - Prob. 18RQCh. 9 - a) Show that every finite subset of a lattice has...Ch. 9 - a) Define a well-ordered set. b) Describe an...Ch. 9 - Let S be the set of all stings of English leers....Ch. 9 - struct a relation on the set {a,b, c, d} that is...Ch. 9 - Show that the relationRonZZdefined by (a, b)R(c,...Ch. 9 - w that a subset of an antisymmetric relation is...Ch. 9 - LetRbe a reflexive relation on a setA. Show...Ch. 9 - Suppose thatR1andR2are reflexive relations on a...Ch. 9 - pose thatR1andR2are reflexive relations on a...Ch. 9 - Suppose that R is a symmetric relation on a set A....Ch. 9 - R1andR2be symmetric relations. IsR1R2also...Ch. 9 - A relationRis called circular ifaRbandbRcimply...Ch. 9 - Show that a primary key in ann-ary relation is a...Ch. 9 - Is the primary key in ann-ary relation also a...Ch. 9 - Show that the reflexive closure of the symmetric...Ch. 9 - Rbe the relation on the set of all mathematicians...Ch. 9 - a) Give an example to show that the transitive...Ch. 9 - a) LetSbe the set of subroutines of a computer...Ch. 9 - pose thatRandSare relations on a set A withRSsuch...Ch. 9 - Show that the symmetric closure of the union of...Ch. 9 - Devise an algorithm, based on the concept of...Ch. 9 - ch of these are equivalence relations on the set...Ch. 9 - How many different equivalence relations with...Ch. 9 - Show that{(x,y)xyQ}is an equivalence relation on...Ch. 9 - pose thatP1={A1,A2,....Am}andP2={B1,B2,....Bm}are...Ch. 9 - Prob. 24SECh. 9 - Prob. 25SECh. 9 - Let P(S) be thesetof all partitions of the set S....Ch. 9 - edule the tasks needed to cook a Chinese meal by...Ch. 9 - Find all chains in the posets with the Hass...Ch. 9 - Prob. 29SECh. 9 - Find an antichain with the greatest number of...Ch. 9 - Show that every maximal chain in a finite poset...Ch. 9 - Prob. 32SECh. 9 - w that in any group ofmn+1people there is either a...Ch. 9 - Prob. 34SECh. 9 - Prob. 35SECh. 9 - Prob. 36SECh. 9 - Prob. 37SECh. 9 - LetRbe a quasi-ordering and let S be the relation...Ch. 9 - w that the following properties hold for all...Ch. 9 - w that ifxandyare elements of a...Ch. 9 - w that ifLis a bounded lattice with upper bound 1...Ch. 9 - w that every finite lattice is bounded. A lattice...Ch. 9 - Give an example of a lattice that is not...Ch. 9 - Show that the lattice(P(S),)whereP(S) is the power...Ch. 9 - the lattice (Z+,)distributive? The complement of...Ch. 9 - Give an example of a finite lattice where at least...Ch. 9 - w that the lattice(P(S))whereP(S)is the power set...Ch. 9 - Show that ifLis a finite distributive lattice,...Ch. 9 - w that the game of Chomp with cookies arranged in...Ch. 9 - w that if(S,)has a greatest elementb,then a...Ch. 9 - Prob. 1CPCh. 9 - Prob. 2CPCh. 9 - Prob. 3CPCh. 9 - Prob. 4CPCh. 9 - Prob. 5CPCh. 9 - Prob. 6CPCh. 9 - Prob. 7CPCh. 9 - Prob. 8CPCh. 9 - Prob. 9CPCh. 9 - Given the matrix representing relation on a finite...Ch. 9 - Prob. 11CPCh. 9 - en the matrix representing a relation on a finite...Ch. 9 - Given the matrix representing a relation on a...Ch. 9 - Prob. 14CPCh. 9 - Prob. 15CPCh. 9 - Prob. 1CAECh. 9 - Prob. 2CAECh. 9 - Prob. 3CAECh. 9 - Prob. 4CAECh. 9 - d the transitive closure of a relation of your...Ch. 9 - pute the number of different equivalence relations...Ch. 9 - Prob. 7CAECh. 9 - Prob. 8CAECh. 9 - Prob. 9CAECh. 9 - Discuss the concept of a fuzzy relation. How are...Ch. 9 - cribe the basic principles of relational...Ch. 9 - Explain how the Apriori algorithm is used to find...Ch. 9 - Describe some applications of association rules in...Ch. 9 - Prob. 5WPCh. 9 - Prob. 6WPCh. 9 - Prob. 7WPCh. 9 - Prob. 8WPCh. 9 - Prob. 9WPCh. 9 - Prob. 10WPCh. 9 - Prob. 11WPCh. 9 - Prob. 12WP
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- Problem: The probability density function of a random variable is given by the exponential distribution Find the probability that f(x) = {0.55e−0.55x 0 < x, O elsewhere} a. the time to observe a particle is more than 200 microseconds. b. the time to observe a particle is less than 10 microseconds.arrow_forwardThe OU process studied in the previous problem is a common model for interest rates. Another common model is the CIR model, which solves the SDE: dX₁ = (a = X₁) dt + σ √X+dWt, - under the condition Xoxo. We cannot solve this SDE explicitly. = (a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler scheme to simulate a trajectory of the CIR process. On a graph, represent both the trajectory of the OU process and the trajectory of the CIR process for the same Brownian path. (b) Repeat the simulation of the CIR process above M times (M large), for a large value of T, and use the result to estimate the long-term expectation and variance of the CIR process. How do they compare to the ones of the OU process? Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000. 1 (c) If you use larger values than above for the parameters, such as the ones in Problem 1, you may encounter errors when implementing the Euler scheme for CIR. Explain why.arrow_forwardRefer to page 1 for a problem involving proving the distributive property of matrix multiplication. Instructions: Provide a detailed proof using matrix definitions and element-wise operations. Show all calculations clearly. Link [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
- Refer to page 30 for a problem requiring solving a nonhomogeneous differential equation using the method of undetermined coefficients. Instructions: Solve step-by-step, including the complementary and particular solutions. Clearly justify each step. Link [https://drive.google.com/file/d/1wKSrun-GlxirS3IZ9qoHazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 5 for a problem requiring finding the critical points of a multivariable function. Instructions: Use partial derivatives and the second partial derivative test to classify the critical points. Provide detailed calculations. Link [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440AZF/view?usp=sharing]arrow_forwardRefer to page 3 for a problem on evaluating limits involving indeterminate forms using L'Hôpital's rule. Instructions: Apply L'Hôpital's rule rigorously. Show all derivatives and justify the steps leading to the solution. Link [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qo Hazb9tC440AZF/view?usp=sharing]arrow_forward
- 3. Let {X} be an autoregressive process of order one, usually written as AR(1). (a) Write down an equation defining X₁ in terms of an autoregression coefficient a and a white noise process {} with variance σ². Explain what the phrase "{} is a white noise process with variance o?" means. (b) Derive expressions for the variance 70 and the autocorrelation function Pk, k 0,1,. of the {X} in terms of o2 and a. Use these expressions to suggest an estimate of a in terms of the sample autocor- relations {k}. (c) Suppose that only every second value of X is observed, resulting in a time series Y X2, t = 1, 2,.... Show that {Y} forms an AR(1) process. Find its autoregression coefficient, say d', and the variance of the underlying white noise process, in terms of a and o². (d) Given a time series data set X1, ..., X256 with sample mean = 9.23 and sample autocorrelations ₁ = -0.6, 2 = 0.36, 3 = -0.22, p = 0.13, 5 = -0.08, estimate the autoregression coefficients a and a' of {X} and {Y}.arrow_forward#8 (a) Find the equation of the tangent line to y = √x+3 at x=6 (b) Find the differential dy at y = √x +3 and evaluate it for x=6 and dx = 0.3arrow_forwardRefer to page 96 for a problem involving the heat equation. Solve the PDE using the method of separation of variables. Derive the solution step-by-step, including the boundary conditions. Instructions: Stick to solving the heat equation. Show all intermediate steps, including separation of variables, solving for eigenvalues, and constructing the solution. Irrelevant explanations are not allowed. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]arrow_forward
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