5. Let \( X \) be the set \( X = \{1, 2, 3\} \), and let \( R \) be the relation defined on \( X \) by \( x_1 \, R \, x_2 \) if \( x_1 + x_2 \neq 3 \).(a) Write \( R \) as a subset of \( X \times X \).(b) Determine if \( R \) is reflexive, symmetric, antisymmetric, and/or transitive.

Answered: Image /qna-images/answer/ab786648-6871-440d-9016-7557620e41a7.jpg

Answered: 5. Let X be the set X = {1,2,3}, and…

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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For the set A = {are, bot, cab, era}, define the relation R on A to be: (x,y) e R if and only if word x and word y share one letter only. For example (are, cab) is in R because those words share one letter only ("a"). But (are, era) is NOT in R because those words do NOT share one letter only, they share 3 letters. Also (bot, era) is NOT in R since they share no letters at all. In cach part, answer yes or no. If yes, then explain why in terms of this specific relation about sharing letters (not just by stating the generic definition of the property). If you say a) Is R reflexive? b) Is R symmetric? c) Is R antisymmetric? d) Is R transitive?
Let R be the relation on the set of ordered pairs of positive integers such that ((a,b), (c,d)) e R if and only if a + d = b + c. Determine whether R is reflexive, symmetric and transitive.
Let R be the relation on a set {1,2,3,4,5} containing ordered pairs (1,1), (1,2),(2,2),(2,5)(3,1),(3,4)(4,1)(4,3)(5,1),(5,3)(5,5). Find the reflexive closure
(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 %D Antisymmetric and transitive. ii. i. Not antisymmetric, and not transitive Antisymmetric, and not transitive Non of these iii. iv.
RUR and RNR symmetric relations on X. are Show that R-Z x Z is an equivalence relation.
let A= {a,b,c,d} and R= {(a,a),(a,c),(b,d),(c,a),(c,c),(d,b)} be a relation on A. What is the inverse of R?
(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
Let R be the relation on N defined by a R b if either a | 2b or b | 2a. Is R an equivalence relation?
Give examples of relations R and S on Z such that (a) Both R and S are not equivalence relations but R+ S is. (b) Both R and S are not partial order relations but R+ S is. Show that if both R and S are equivalences relations, then R+S is not an equivalence relation.
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c,d)) = R if and only if ad = bc. Show R is an equivalence relation.

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5. Let X be the set X = {1,2,3}, and let R be the relation defined on X by x₁ Rx₂ if x1 + x2 3. (a) Write R as a subset of X X X. (b) Determine if R is reflexive, symmetric, antisymmetric, and/or transitive.