5.For each of the following partial orders, draw a Hasse diagram, find themaximal and minimal elements, and determine whether the partial order is a total order.a. Ris a partial order on set {2,4, 5, 7,10, 25,40} such that (x, y) E R if and only if x > y.b. R is a partial order on set {2,4, 5,7,10,25, 40} such that (x, y) E R if and only if x is amultiple of y. Hint: An integer x is a multiple of an integer y if and only if x is divisible by y.

Answered: Image /qna-images/answer/1c20abdc-f296-4e5e-933a-fadd1c1ca343.jpg

Answered: 5. For each of the following partial…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. List all possible arrangements of the four letters m, a, t, and h. Let C₁ be the collection of the arrangements in which h is in the last position. Let C₂ be the collection of the arrangements in which m is in the first position. Find the union and the intersection of C₁ and C₂.
(6) 13. Let S = {a,b,c,d}. Let r be the partial order {(a,a),(b,b), (c,c),(d,d),(b,d),(c,a)}. Give the Hasse diagram for r.
Draw the Hasse diagram for the following partial order.
Draw the Hasse diagram for the following partial order. Domain: {3, 5, 6, 7, 10, 14, 20, 30, 60}. x
Please answer A to C.
Let R = {(a, a),(a, b), (b, b), (a, c), (c, c)} be a partial order relation on Σ = {a,b,c}. Let be the corresponding lexicographic order on Σ*, the set of all finite strings over Σ. Which of the following is true? Select one: O a. abbac abb O b. abbac O c. O d. abbac abbaqc O e. abbaaacc abbaab be ba abbab
Which answer is correct?
Let A = {0,1,2} and the partial order relation R be defined on the set of A × A as follows: For V(a, b), (c, d) E A × A, (a,b)R(c,d) → a < c and b < d |(a) Draw the Hasse diagram for R. (b) Write the minimal, maximal, least and greatest elements of R.
For any partial order there may be a number of linear orders that are consistent. For example,for the partial order (P (A), ⊂) with A = {1, 2, 3}, the ordering∅ ≺ {1} ≺ {2} ≺ {1, 2} ≺ {3} ≺ {1, 3} ≺ {2, 3} ≺ {1, 2, 3}does not violate the partial order, i.e. if a ≺ b → b ̸ ⊆ a. Find all the consistent linear orders for(P (A), ⊂) when A = {1, 2, 3, 4}.
Part 2. Each relation given below is a partial order. Draw the Hasse diagram for the partial order. (a) The domain is {3, 5, 6, 7, 10, 14, 20, 30, 60}. x ≤ y if x evenly divides y. 1 (b) The domain is {a, b, c, d, e, f}. The relation is the set: {(b, e), (b, d), (c, a), (c, f), (a, f), (a, a), (b, b), (c, c), (d, d), (e, e), (ƒ, f)}
4. Let A := {1, 2, 3, 4} := B. (a) What is the cartesian product of A and B. (b) Find S := {(a, b)|f € Z}. (d) Is S a Partial Order? Justify your answer.
Consider an infinite set of undefined elements S and the undefined relation R that satisfies the following axioms: Axiom 1. If a, b e S and a Rb, then a #b. Axiom 2. If a, b, c e S, a Rb, and bRc, then a Rc. (i) Show that an interpretation with S as the set of integers and a Rb as "a is less than b" is a model for the system. (ii) Would S as the set of integers and a Rb interpreted as "a is greater than b" also be a model? (iii) Are the models in parts (a) and (b) isomorphic? (iv) Would S as the set of real numbers and a Rb interpreted as "a is less than b" be another model? (v) Is the model in part (d) isomorphic to the model in part (a)?

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