1. Let A = {1,3,5}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule (X,Y) ∈ R ⇔ (X ∩ Y = Ø) ∧ (X ∪ Y = A). (a) Prove or find a counterexample for the following properties and based on the findings from the properties, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.: antisymmetry asymmetry symmetry irreflexivity reflexivity transitivity uniqueness

1. Let A = {1,3,5}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule (X,Y) ∈ R ⇔ (X ∩ Y = Ø) ∧ (X ∪ Y = A). (a) Prove or find a counterexample for the following properties and based on the findings from the properties, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.: antisymmetry asymmetry symmetry irreflexivity reflexivity transitivity uniqueness

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. We are given the sets S and V: S = {Anne, Theo} V {studies, runs, meditates} = (a) Find the Cartesian Product of S and V: S x V (b) A relation R. is a subset of the Cartesian product. Create such a subset RC (S X V) with three elements.
(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.
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7. List the coordinate pairs needed in order to turn the following relations into equivalence relations. AR
Let ? = {1, 2, 3, 4, 5, 6, 7} and define a relation ? ⊆ ? × ? by? = {(1,1), (1,2), (1,3), (2,3), (7,1), (7,2), (7,3), (4,5), (4,6), (5,6), (5,5)}Determine (yes or no) whether ? has the following properties (no justification is required).
3. Let R be a relation defined on the set A = N × N where (a, b) R (c, d) if að = cd. Show R is an equivalence relation. Then, list the elements of the equivalence class [(9, 2)].
Provide right answer with complete solution Answer letter C only.
For universe S = R², define relation ~= as: (a,b) ~= (c,d) < 2a - b = 2c - d Prove that ~= is an equivalence relation on R², or find a counterexample.
3. Define a relation on the real numbers as follows: x Ry if and only if x² + y² = 1. Determine if R is reflexive, symmetric, and/or transitive. For each property, give sufficient justification for your answer.
1. Let A = {2,4,6,8}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule (X,Y) ∈ R ⇔ (X ∩ Y = Ø) ∧ (X ∪ Y = A). (a) Prove or find a counterexample for the following properties and based on the findings from the properties, conclude whether R is a graph, partially ordered set, equivalence relation, or a function (several answers may hold simultaneously). Analyze the relation R.: antisymmetry asymmetry symmetry irreflexivity reflexivity transitivity uniqueness