Consider an infinite set of undefined elements S and the undefinedrelation 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 anda 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 isgreater 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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Answered: Consider an infinite set of undefined…

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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When |S| denotes the number of elements of the set S, and S = {x/1 ≤ x ≤ 100, x € integer } There are three subsets of S, namely sets A, B and C. It is known |A|=(10+k), |B|= (7+2k), |C|= (8+2k), |A∩B|=(2+k), |A∩C|=(3 +k), |B∩C|=2k, |A∩B∩C|=k and |(AUBUC)'|=40 Draw the Venn diagram, determine the value of k ? Determine the number of elements of each set A, B and C?
This question refers to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from a set A to a set B, RUS= {(x, y) E A × B| (x, y) ERor (x, y) E S} RO S= {(x, y) E A x B | (x, y) ERand (x, y) E S}. Let A = {-4, 4, 7, 9} and B = {4, 7}, and define relations R and S from A to B as follows. For every (x, y) E Ax B, x R y x| = [y] and x Sy x - y is even. Which ordered pairs are in A x B, R, S, R U S, and RN S? (Use set-roster notation to enter your answers.) Ax B = R = S = RUS = ROS =
This question refers to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from a set A to a set B, RUS = {(x, y) E AXB I (x, y) ER or (x, y) E S} RNS = {(x, y)E AXB| (x, y) ER and (x, y) Es}. Let A = {-2, 2, 3, 5} and B = {2, 3} and define relations R and S from A to B as follows: For every (x, y) E Ax B, x R y - |x| = lyl and xsyex - y is even. Using set-roster notation, state explicitly which ordered pairs are in A x B, R, S, RU S, and RN s. (Enter your answers as comma-separated lists of ordered pairs.) AXB = R = S = RUS = ROS =
b. Identify the systems from task (2a) that demonstrate closure, i.e., the result of the operation always produces an element in the set.c. Identify the systems from task (2a) that have an identity element.
17 Part 4 of 4 Required information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the following relations: R₁ = {(a, b) € R² | a> b), the greater than relation R₂ = {(a, b) ER² | a2 b), the greater than or equal to relation R3 = {(a, b) ER² | a< b), the less than relation R₁ = {(a, b) ER² | a ≤ b), the less than or equal to relation R₁ = {(a, b) = R² | a = b), the equal to relation R₁ = {(a, b) € R² | a‡ b), the unequal to relation For these relations on the set of real numbers, find R₁ R6. Multiple Choice O O о R5 R² R6
14 Part 1 of 4 Required information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the following relations: R₁ = {(a, b) = R² | a> b), the greater than relation R₂ = {(a, b) € R2 R3 = {(a, b) € R² | a < b), the less than relation R₁ = [(a, b) ER² | a ≤ b), the less than or equal to relation R5 = {(a, b) € R² | a= b), the equal to relation R₁ = {(a, b) = R² | a b), the unequal to relation For these relations on the set of real numbers, find R₂0 R2. Multiple Choice R4 R3 I az b), the greater than or equal to relation R2 R₁

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