**Problem 6: Relations on a Set**Let \( S \) be a set, and suppose \(\sim\) is a relation on \( S \). For each of the following, provide a counterexample:(a) If \(\sim\) is reflexive and symmetric, is it transitive?(b) If \(\sim\) is reflexive and transitive, is it symmetric?(c) If \(\sim\) is symmetric and transitive, is it reflexive?

Let us solve the given problems on relation in the next steps.

Answered: 6. Let S be a set and suppose is a…

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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I have the following questions: " Let R = ∅ be the empty relation on a nonempty set A. Which of the properties reflexive, symmetric andtransitive does R possess?" and the following answer is given: "Let set A is nonempty set and consider relation R given by R=∅. Then we can conclude that the relation R is symmetric because {∅,∅}∈R. The only element of R is ∅ so relation is transitive. Since A is a nonempty set then relation R is not reflexible." How can it be any propert when the set is empty? don't you need pairs to decide which property the set holds?
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1. Consider the relation < on the set {1,2, 3, 4}. Write down this relation as a set of ordered pairs in roster form.
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Let - be a relation on N defined as: a ~b if a < (b – 1). (i) Is ~ reflexive? Prove your answer. (ii) Is ~ (iii) Is ~ transitive? Prove your answer. symmetric? Prove your answer.
Let S be a nonempty set. Consider the relation ~ on P(S) defined by A ~ B if AUBC = S. Is the relation an equivalence relation? Explain. If yes, identify (no need to justify) the equivalence class of
Consider the following two statements: (1) If a relation on a nonempty set is reflexive, then it is not irreflexive. (2) If a relation on a nonempty set is not reflexive, then it is irreflexive O Neither (1) nor (2) is always true. O Only (1) is always true. O Only (2) is always true. Both (1) and (2) are always true.
This question confuses me, I dont get what is being asked? How do I approach this? "Let A be a nonempty set and let B be a subset of the power set P(A) of A. Define a relation R from A to B byxRY if x ∈ Y. Give an example of two sets A and B that illustrate this. What is R for these two sets?"
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6. Let S be a set and suppose is a relation on S. For each of the following, provide a counterexample. (a) If (b) If (c) If ~ is reflexive and symmetric, is it transitive? is reflexive and transitive, is it symmetric? 1 is symmetric and transitive, is it reflexive?