**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

See similar textbooks

Related questions

Q: Given set B as B = {0,1,2,3,4}. Suppose set A is the relation and defined as a. A = {(x, y) E Z | xy…

Q: 2. For each of the following relations, indicate whether it is a partial order, a strict order, or…

Q: A playground consists of several structures, some of which are connected by bridges. The ground is…

Q: Determine whether the relation with the directed graph is an equivalence relation:

Q: In each item below, verify that ~ is an equivalence relation on the given set. Let S = Z × N. Define…

A: Since you have asked multiple questions, we will solve the Question Q (a) and (b) for you. If…

Q: Another familiar relation is the "greater than or equal to" (2) relation. Let A be the set of…

A: Given relation is "Greater than or equal to"

Q: d. Ris a relation on Z* such that (x, y) e R if and only if there is a positive integer n such that…

Q: 8. Draw the directed graph of the binary relation S on B = { a, b, c, d, e} by S = {(a, b),(b,…

Q: Let R be a relation on the set A = {0, 1, 4, 8} defined by xRy if and only if x + y < 8. (i) List…

Q: Answer the following question accordingly:

A: Given set is B={1,2,3,4}

Q: . A) Write the ordered pairs, graphical representation, and the relation matrix, for the Relation R…

Q: Let A = {2, 3, 4, 6, 7, 9} and define a relation R on A as follows: For all x, y e A , x R y = 3…

A: Given A = {2,3,4,6,7,9} xRy ⇔ 3 divides (x-y) Therefore 1) R = (2,2) , (3,3) , (4,4) , (6,6) ,…

Q: 7. Let S = {1,2,3, ...,8}. Let R be the relation on S defined by the following: %3D xRy + xy is a…

Q: 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. draw a…

Q: c. Ris a relation on Z such that (x, y) e R if and only if xy 2 0. d. Ris a relation on Z* such that…

Q: For each of the following relations, determine whether the relation is: Transitive. A partial order.…

A: Answer:Explanation:Reflexive: A relation on a set is reflexive if every element in the set is…

Q: 7. Let A=(1:2:3; 4) and R be the relation on (A) List the ordered pairs in the relation R. (B) Find…

A: Since you have posted multiple questions we will provide the solution only to the first question as…

Q: Let R be a relation over a set A = {a,b,c,d}. What kind of order is R? Note: Select the order that…

Q: Create (a) the ordered pairs, (b) graphical representation, and (c) the relation matrix, for the…

A: Given a relation R on the set B as "is factor of" B=1,2,3,6. a The ordered pair is…

Q: Decide whether each of the following relations is an equivalence relation or not. {(a, b) | gpa of…

A: To prove equivalence relation, we have to prove that (i) the relation is reflexive, (ii) the…

Q: lation. Justify why your answers are corre R is a relation on the set of all people sue R is a…

A: EQUIVALENCE RELATION: If a relation R on a set A satisfies the following three criteria, it is…

Q: Let A = {a, b, c, d} and define a relation R on A. The directed graph of the relation is shown here.…

A: A relation R on a set A with elements a, b and c is said to be in a transitive relationship, then if…

Q: The following definitions are used: a relation on a set A is defined to be irreflexive it, and only…

A: The given set is A=01,2,3 The given relation is R=0,0,0,1,0,2,1,2. (i) Using the definitions of…

Q: Let A = {a, b, c, d} and define a relation R on A. The directed graph of the relation is shown here.…

A: We have: The relation is: R=a,c,b,a,c,d,d,b

Q: 6. Let A {1,2, 3, 4, 5} and define a relation on A as follows: For all x, y E A, %3D xRy + x+ y is…

Q: draw the graph of this relation. Search this relation

Q: Let R be a relation from A to B and defined "x is at least twice as big as y." Write R as a set of…

Q: Suppose that P is a unary relation symbol, Q is a binary relation symbol, f is a unary function…

A: Note: According to bartleby we have to answer only first three subparts please upload the question…

Q: Let X = {0, 1, 2, 3, 4}. Draw a relation R on X such that xRy if x + y = 4. a. Draw the graph…

Q: Exercise 17.2.18. Find binary relations on {1,2,3} that meet each of the following conditions.…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: Let A and B be any two sets. A binary relation from A to B is i. CAXB ii. SANB CAUB Non of these…

A: Firstly we define binary relation from set A to set B, and then conclude the answer.

Q: prove why each relation has or does not have the properties: reflexive, symmetric, anti-symmetric,…

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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Relations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

For the following directed graph, do the following: 1. give (in set notation, e.g. { (a,b), (c,x), ...}) the set that corresponds to the graph expressed as a relation (set of ordered pairs). (8 pts) 2. is the relation reflexive? (2 pts) Why or why not? 3. is the relation symmetric? (2 pts) Why or why not? 4. is the relation transitive? (4 pts) Why or why not? a b d e
4. Define the relation r on S = {A, B, C, D} by r ={ (A,B), A,D), (B,A), (B,C), (C,A), (C,B), (D,B)}. Give the relation digraph for r.
Please do Exercise 17.1.8 part A,B,C and please show step by step and explain Hint A: There are two. Hint B: There are four Hint C: There are four.
d. The business major has attended FSU for 2 yea 4. Let C = (0, 1, 2} and D = {2,4,6, 8) and define a relation R from A to Bas follows: for all (x,y) E AX B, (x, y) ER %3D %3D is an integer. a. Write A x B as a set of ordered pairs. b. Is 1 R2? Is 2 R8? Is 1 R 8? Is 2 R6? c. Write R as a set of ordered d. Draw an arrow diagram for R. means that Y+2 pairs.
1. Let A = {2, 3, 4, 5, 6} and B = {10, 11, 12, 13}. Define a relation R on (A x B) as follows: a Rb if and only if gcd(a, b) = 1. a) Write out the elements of the relation R explicitly (remember, R is the set of ordered pairs from (A x B) that satisfy the given property). b) Write out the elements of the relation R1 explicitly.
A conjunt "C" groups 8 topics from the calculus course (a,b,c,d,e,f,g,h) trying to identify the liking for the course, a relation T in CxC is generated such that (x,y) ∈ T if and only if "the topic y is liked as much or more than the topic x". The answers are set out in the following Hasse diagram: (image) What is the favorite topic for the students? and What is the topic that students like the least?
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?
1. Consider the relation < on the set {1,2, 3, 4}. Write down this relation as a set of ordered pairs in roster form.
The domain of a relation A is the set of integers. x is related to y under relation A if x = y + 2. The relation B is the reflexive closure of the transitive closure of A.(a) Is A partial order? Is A total order? Justify both your answers. (b) Is B partial order? Is B total order? Justify both your answers.
MTHH039059 Matching Match each relation in questions 1-3 with one of the following models. WORDBANK < A. 21² -2 O B. 1 B 2 2 21- 1. Which graph models the relation {(1, 2), (-1, -2), (2, -1)}? hp -2 0 Which graph models the relation {(-1, 2), (-2, 1), (-1, -2)}? W
Given a set A = {1, 2, 3, 4}. Define a relation R on A as: R = {(x, y) | x - y is even} List the ordered pairs of the relation R. Is the relation R reflexive, symmetric, and transitive? Justify your answers.
20 - Determine whether the relation with the directed graph shown is an equivalence relation ? a b d a) O Yes b) O No