/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 \((A, R),\) where \(A=\\{a, b, c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 29

\((A, R),\) where \(A=\\{a, b, c\\}\) and \(R=\\{(a, a),(a, b),(b, b),(b, c),(c, c)\\}\)

Short Answer

Expert verified
The relation R on set A is reflexive since it contains all the ordered pairs (x, x) for all x in the set A. However, it is not symmetric, as the pair (a, b) is in R, but (b, a) is not. Additionally, R is not transitive, as we found a counterexample with pairs (a, b) and (b, c) being in R, but (a, c) not present. Therefore, R is only reflexive.

Step by step solution

01

Determine if the relation is reflexive

For a relation to be reflexive, it must contain all the ordered pairs (x, x) for all x in the set A. In other words, it should contain (a, a), (b, b), and (c, c). Looking at the given relation R, we see that it does contain these ordered pairs, so R is reflexive.
02

Determine if the relation is symmetric

For a relation to be symmetric, it must satisfy the condition that if (x, y) is in R, then (y, x) must also be in R. To check this, we look at the ordered pairs in R: (a, a) 鈥� (a, a) is also in R (Reflexive pairs are always symmetric), (a, b) 鈥� We need to check if (b, a) is in R, and it's not in R, so R is not symmetric. Since we found a counterexample, there is no need to check the remaining pairs. Therefore, R is not symmetric.
03

Determine if the relation is transitive

For a relation to be transitive, it must satisfy the condition that if (x, y) and (y, z) are in R, then (x, z) must also be in R. Now, we check this for R: (a, a), (a, b) 鈥� We have x=a, y=a, and z=b. Since (a, b) is in R, the transitivity holds for these pairs. (a, b), (b, b) 鈥� We have x=a, y=b, and z=b. Since (a, b) is in R, the transitivity holds for these pairs. (a, b), (b, c) 鈥� We have x=a, y=b, and z=c. Since (a, c) is not in R, the transitivity doesn't hold for these pairs. We found a counterexample showing that the relation is not transitive. Therefore, R is not transitive. So, according to our analysis, the relation R on set A is reflexive, but not symmetric or transitive.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflexive Relation
A reflexive relation on a set means that every element in the set is related to itself. This is an important concept in discrete mathematics.
In simpler terms, if you have a set \( A \) which contains elements \( a, b, \) and \( c \), a relation \( R \) on \( A \) is said to be reflexive if every element is paired with itself within the relation.

To illustrate, consider:
  • (a, a)
  • (b, b)
  • (c, c)
These are the ordered pairs required for a reflexive relation on the set \( A \). Each element is reflecting onto itself, like looking in a mirror.

For the given relation \( R = \{(a, a), (a, b), (b, b), (b, c), (c, c)\} \):
  • (a, a) is present
  • (b, b) is present
  • (c, c) is present
Since all needed pairs are included, we conclude the relation \( R \) is reflexive.
Symmetric Relation
Symmetry in relations refers to the idea that if an element is related to another, then the reverse should also be true. In a symmetric relation, for every pair (x, y), there must also be an equivalent pair (y, x).

Looking at the relation \( R = \{(a, a), (a, b), (b, b), (b, c), (c, c)\}\), let's evaluate its symmetry:
  • (a, a) implies (a, a) 鈥� no issue here as it's reflexive too.
  • (a, b) would require (b, a) to also be present for symmetry, but (b, a) is not in \( R \).
This missing pair breaks the symmetry condition. If we find even one pair which doesn鈥檛 fulfill the symmetric condition, the whole relation ceases to be symmetric.
Therefore, the relation \( R \) is not symmetric.
Transitive Relation
Transitive relations concern the flow of connections: if a relation links \( x \) to \( y \), and \( y \) to \( z \), it should also link \( x \) to \( z \).
To verify if \( R \) is transitive, examine:
  • (a, a) with (a, b) leads us back to (a, b), which is fine.
  • (a, b) with (b, b) also leads us correctly to (a, b).
  • (a, b) with (b, c) should result in (a, c), but (a, c) is absent in \( R \).
Just one missing link, such as the absence of (a, c), means \( R \) is not transitive. The presence of a counterexample like this one reveals that the condition of transitivity is not met.
Consequently, since we identified this inconsistency, the relation \( R \) is not transitive.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

When will a relation \(R\) on a set \(A\) be both symmetric and antisymmetric?

A relation \(R\) on a set \(A\) is irreflexive if no element of \(A\) is related to itself, that is, if \((a, a) \notin R\) for every \(a \in A .\) Determine if each relation is irreflexive. The less-than relation on \(\mathbb{R}\).

Let \(A\) denote the set of words of length \(\leq 3\) over the binary alphabet. The relation \(R,\) defined on \(A\) by \(x R y\) if \(x\) is a prefix of \(y,\) is a partial order. Draw a Hasse diagram for the poset \((A, R)\).

The number of partitions of a set with size \(n\) is given by \(\sum_{r=1}^{n} S(n, r),\) where \(S(n, r)\) denotes a Stirling number of the second kind. Compute the number of partitions of a set with the given size. Five

Determine if each is a partial order on \(\\{a, b, c\\}\). $$\\{(a, a),(b, b),(b, c),(c, c)\\}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.