**Relation Properties and Directed Graphs**In problems 1-8, several relations are defined on the set $ A = \{0, 1, 2, 3\} $. For each relation, perform the following tasks:a. **Draw the Directed Graph**: Construct a directed graph representing the relation. The vertices will be the elements of the set $ A $, and directed edges will represent the pairs in the relation.b. **Determine Reflexivity**: Check if the relation is reflexive. A relation is reflexive if every element in the set is related to itself, meaning each vertex should have a loop.c. **Determine Symmetry**: Examine if the relation is symmetric. A relation is symmetric if for every pair (a, b) in the relation, the pair (b, a) is also present. This means that for every directed edge from a to b, there should be an edge from b to a.d. **Determine Transitivity**: Investigate if the relation is transitive. A relation is transitive if whenever there is a directed edge from a to b and from b to c, there is also a direct edge from a to c.For each of these properties, if the relation does not satisfy one, provide a counterexample to illustrate why. This involves showing specific elements that violate the property.These exercises aim to deepen your understanding of the properties of relations and how they can be represented visually through directed graphs. **2.** $ R_2 = \{(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 2), (2, 3)\} $In this mathematical notation, $ R_2 $ is defined as a set of ordered pairs. Each pair is a binary relation consisting of two elements. This type of relation can be used to describe mappings or connections between elements in a set.- **(0, 0)**: A relation where 0 is related to itself.- **(0, 1)**: A relation where 0 is related to 1.- **(1, 1)**: A relation where 1 is related to itself.- **(1, 2)**: A relation where 1 is related to 2.- **(2, 2)**: A relation where 2 is related to itself. Note that this pair appears twice, indicating redundancy but no change in interpretation.- **(2, 3)**: A relation where 2 is related to 3.This set of relations could be used to represent various mathematical concepts, such as a graph, a transition table in automata theory, or simply a collection of mapped pairs in a function or matrix.

Given: The relation defined on the set A=0,1,2,3 is R2=0,0,0,1,1,1,1,2,2,2,2,3 We want to determine…

In 1-8, a number of relations are defined on the set A = {0, 1, 2, 3}. For each relation: a. Draw the directed graph. b. Determine whether the relation is reflexive. c. Determine whether the relation is symmetric. d. Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties.

In 1-8, a number of relations are defined on the set A = {0, 1, 2, 3}. For each relation: a. Draw the directed graph. b. Determine whether the relation is reflexive. c. Determine whether the relation is symmetric. d. Determine whether the relation is transitive. Give a counterexample in each case in which the relation does not satisfy one of the properties.

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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Question

**Relation Properties and Directed Graphs**

In problems 1-8, several relations are defined on the set $ A = \{0, 1, 2, 3\} $. For each relation, perform the following tasks:

a. **Draw the Directed Graph**: Construct a directed graph representing the relation. The vertices will be the elements of the set $ A $, and directed edges will represent the pairs in the relation.

b. **Determine Reflexivity**: Check if the relation is reflexive. A relation is reflexive if every element in the set is related to itself, meaning each vertex should have a loop.

c. **Determine Symmetry**: Examine if the relation is symmetric. A relation is symmetric if for every pair (a, b) in the relation, the pair (b, a) is also present. This means that for every directed edge from a to b, there should be an edge from b to a.

d. **Determine Transitivity**: Investigate if the relation is transitive. A relation is transitive if whenever there is a directed edge from a to b and from b to c, there is also a direct edge from a to c.

For each of these properties, if the relation does not satisfy one, provide a counterexample to illustrate why. This involves showing specific elements that violate the property.

These exercises aim to deepen your understanding of the properties of relations and how they can be represented visually through directed graphs.

**2.** $ R_2 = \{(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 2), (2, 3)\} $

In this mathematical notation, $ R_2 $ is defined as a set of ordered pairs. Each pair is a binary relation consisting of two elements. This type of relation can be used to describe mappings or connections between elements in a set.

- **(0, 0)**: A relation where 0 is related to itself.
- **(0, 1)**: A relation where 0 is related to 1.
- **(1, 1)**: A relation where 1 is related to itself.
- **(1, 2)**: A relation where 1 is related to 2.
- **(2, 2)**: A relation where 2 is related to itself. Note that this pair appears twice, indicating redundancy but no change in interpretation.
- **(2, 3)**: A relation where 2 is related to 3.

This set of relations could be used to represent various mathematical concepts, such as a graph, a transition table in automata theory, or simply a collection of mapped pairs in a function or matrix.

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