Let S = {2, 4, 6, 8, 10, 13, 15}, and let be a relation on S defined byx≤y if x = y or 2x ≤y. Then (S, ) is a poset. [You need not prove this.](a) Find all maximal and all minimal elements of S with respect to <.(b) Find a subset of S of size three such that it has no upper bound in S.(c) Find the sets of lower bounds and upper bounds for Y = {2,6} in S.(d) Determine whether or not the subset {2,8, 15} of S is totally orderedwith respect to. Justify your answer.

The given set is S=2,4,6,8,10,13,15 The relations is defined as a≺b if a=b or 2a≤b. By the relation…

Answered: be a relation on S defined by Let S =…

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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All of the given relations are onto. True or false
Consider the family of sets S= [{1,5, 6}, {2,4, 6), (4, 5, 6), {2, 5, 6}, {3,4, 5}, {1, 4, 6}]. (1.1) and let Y - (1,3, 4, 5}. Define the relation Ron S by (A, B) ER = A\Y = B\Y, where A, B are arbitrary elements of S. (1) (a) Since for every A ES A\Y = \Y, where ? = B ,the relation R (here and below in (b-d), please type is or is not in the input field below) is reflexive. (b) Given any A, BES, we have that A\Y = B\Y = B\Y= \Y where and hence the relation R is symmetric. (c) For every A, B, CES, (A\Y = B\Y) and (B\Y = C\Y) implies A\Y = \Y, where al= ,and hence the relation R is transitive. (d) Accordingly, by (a,b,c), the relation R is an equivalence relation on S. (11) Find the matrix MR of the relation R (please enter the matrix row-by-row in the six input fields below, when entering each row, seperate the entries by single spaces; understandably, the i-th row/column of the matrix MR must correspond to the i-th set A, in the list of the elements of S given in Eq. (1.1) above): (I)…
Let R be the relation defined on P({1,., 100}) by ARB if and only if |A n B| is even. Is R reflexive? Is R symmetric? Is R anti-symmetric? Is R transitive?
Assume R1 and R2 are reflective relations on a set A. Prove or disprove each of the following statements. (a) R1 U R2 is reflexive. (b) Rị N R2 is reflexive.
2. For each of these relations on the set {1, 2, 3, 4), decide whether it is reflexive, whether it is symmetric, whether it is antisymmetric, and whether it is transitive. a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)} b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4,4)} c) {(2, 4), (4, 2)} d) {(1, 2), (2, 3), (3, 4)} e) {(1, 1), (2, 2), (3, 3), (4,4)} f) {(1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4)}
Let T be the set {w = {0, 1}* ||w| ≤ 4}. Let R be the equivalence relation defined on T as follows: R = {(x, y) | x ≤T, yɛT, no(x) = = no(y)}, where no(r) represents the number of zeroes in the string x, and no(y) represents the number of zeroes in the string y. For example, (1011, 01) is a pair in R because the two strings 1011 and 01 have the same number of zeroes as each other. Every element in the set will appear in exactly one equivalence class and will be related to all elements in its class and not related to any elements outside of its class. What are the equivalence classes of T created by the relation R?
In the set A = {1, 2, 3, 4} is defined a relation R by %3D R = {(1,1), (1, 3), (3, 4), (4, 3)}. Indicate all of the following pairs that belong to the reflexive closure of R. O (1, 1) (1, 2) O (1, 3) O (1,4) O (2, 1) O (2, 2) о (2, 3) о (2,4) O (3, 1) (3, 2) O (3, 3) (3, 4) O (4, 1) O (4, 2) O (4, 3) O (4, 4)
A. Let R be a relation on set A. A={x : x ∊ J and 0 ≤ x ≤ 12}, and R={(a,b) : a - b is multiple of 4 } . Represent this relation by a matrix. B. Using induction, prove that n2 + n is divisible by 2, where n ∊ N
(b) Let R = {(1, 1), (1, 2),(2, 1), (2, 2),(3, 3)}. be a relation on a set A = {1,2,3} , the R is Antisymmetric and transitive. ii. i. Not antisymmetric, and not transitive Antisymmetric, and not transitive iv. iii. Non of these

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