/*! 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 4 List the elements in each relati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 4

List the elements in each relation from \(A=\\{1,3,5\\}\) to \(B=\\{2,4,8\\}\). \(\\{(a, b) | a \text { is a factor of } b\\}\)

Short Answer

Expert verified
The elements in the relation are \(\{(1,2), (1,4), (1,8)\}\).

Step by step solution

01

Create a list of ordered pairs

List all possible ordered pairs (a, b) containing one element from A and one element from B. For this, we have A={1,3,5} and B={2,4,8}. The possible ordered pairs are: (1,2), (1,4), (1,8), (3,2), (3,4), (3,8), (5,2), (5,4), (5,8)
02

Identify ordered pairs satisfying the condition

Now, for each ordered pair, we determine whether the element from set A is a factor of the element in set B. In other words, we check if b is divisible by a without leaving a remainder. - (1,2): 1 is a factor of 2 - (1,4): 1 is a factor of 4 - (1,8): 1 is a factor of 8 - (3,2): 3 is not a factor of 2 - (3,4): 3 is not a factor of 4 - (3,8): 3 is not a factor of 8 - (5,2): 5 is not a factor of 2 - (5,4): 5 is not a factor of 4 - (5,8): 5 is not a factor of 8
03

List the elements in the relation

From step 2, we can see that only the ordered pairs (1,2), (1,4), and (1,8) satisfy the condition that a is a factor of b. Therefore, the elements in the relation are: {(1,2), (1,4), (1,8)}

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.

Set Theory
Set theory is a fundamental part of mathematics that deals with collections of objects, known as sets. In this exercise, two sets are defined: set \(A = \{1, 3, 5\}\) and set \(B = \{2, 4, 8\}\).
Sets are usually denoted by curly brackets \(\{\}\) and contain a collection of distinct items, called elements or members.
The elements within a set do not need to be in any particular order, just like our example sets \(A\) and \(B\), where the elements are integers.
  • Set \(A\) and set \(B\) are both finite because they have a countable number of elements.
  • In this context, the goal is to evaluate relations between these sets, involving factors and divisibility.
Understanding basic set notation and properties lays the groundwork for more complex topics like ordered pairs and relations.
Ordered Pairs
An ordered pair is a mathematical tool used to pair two objects, typically in the form \((a, b)\). This is done where \(a\) is a member from the first set and \(b\) from the second set.
In our context, the sets \(A\) and \(B\) help to form these pairs by selecting one element from each set.
  • For example, \((1, 2)\) is an ordered pair where 1 is from set \(A\) and 2 is from set \(B\).
  • The importance of order here is crucial; \((a, b)\) is different from \((b, a)\).
Ordered pairs form the backbone of relations and functions by connecting elements from different sets in a structured manner.
Divisibility
Divisibility refers to the ability of one integer to be divided by another without leaving a remainder.
In mathematics, we say that a number \(a\) is a factor of another number \(b\) if and only if there exists an integer \(c\) such that \(b = a \cdot c\).
In this exercise, we focus on whether elements of set \(A\) are factors of elements in set \(B\).
  • For instance, 1 is a factor of all numbers because any number divided by 1 is itself.
  • Conversely, numbers like 3 or 5 are not factors of 2, 4, or 8 as they do not divide these numbers exactly.
The pairs \((1, 2)\), \((1, 4)\), and \((1, 8)\) fit the criteria because the second elements in these pairs are divisible by the first.

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

Let \(a, b, c, d, m \in \mathbf{Z}\) with \(m \geq 2 .\) Prove each. If \(a \equiv r(\bmod m)\) and \(0 \leq r

Every bank check has an 8.digit identification number \(d_{1} d_{2} \ldots d_{8}\) followed by a check digit d given by \(d=\left(d_{1} d_{2}, \ldots, d_{8}\right)\) \((7,3,9,7,3,9,7,3)\) mod \(10,\) where \(\left(x_{1}, x_{2}, \ldots, x_{n}\right) \cdot\left(y_{1}, y_{2}, \ldots, y_{n}\right)=\) $$\sum_{i=1}^{n} x_{i} y_{i}$$ (It is the dot product of the two n-tuples.) Compute the check digit for \(17,761,976\)

Find three ordered pairs of positive integers that precede the pair (2,3) in lexicographic order.

Give an example of a relation on \(\\{a, b, c\\}\) that is: Neither symmetric nor antisymmetric.

Give an example of a relation on \(\\{a, b, c\\}\) that is: Antisymmetric, but not symmetric.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.