/*! 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 18 For sets \(A, B, C \subseteq Q\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 18

For sets \(A, B, C \subseteq Q\), prove or disprove (with a counterexample), the following: If \(A \subseteq B, B \nsubseteq C\), then \(A \nsubseteq C\).

Short Answer

Expert verified
By providing a counterexample with \(A = \{1\}, B = \{1, 2\}, C = \{1, 3\}\), where \(A\) is a subset of both \(B\) and \(C\), but \(B\) is not a subset of \(C\), we disprove the statement. Therefore, the statement 'if \(A \subseteq B\) and \(B \nsubseteq C\), then \(A \nsubseteq C\)' is not always true in the context of set theory.

Step by step solution

01

Understand the problem

The problem is about sets and subset relationships. In set theory, a set \(A\) is a subset of \(B\) if all elements of \(A\) also belong to \(B\). The task is to analyze whether, given \(A \subseteq B\) and \(B \nsubseteq C\), it necessarily means that \(A \nsubseteq C\).
02

Disprove the statement by providing a counterexample

To disprove a mathematical statement, it's enough to show a single counterexample that violates the statement. Let's consider three sets: \(A = \{1\}\), \(B = \{1, 2\}\), and \(C = \{1, 3\}\). In this case, \(A\) is indeed a subset of \(B\), as every element of \(A\) is in \(B\). At the same time, \(B\) is not a subset of \(C\), as \(2\) is in \(B\) but not in \(C\). However, \(A\) is still a subset of \(C\), as \(1\) is both in \(A\) and in \(C\). So this example contradicts the statement 'if \(A \subseteq B\) and \(B \nsubseteq C\), then \(A \nsubseteq C\)'.
03

Formulate the conclusion

By showing an explicit counterexample, the initial statement can be disproved. Even if a set \(A\) is a subset of \(B\) and \(B\) is not a subset of \(C\), it does not necessarily mean that \(A\) is not a subset of \(C\). It depends on the actual elements in each set, so the statement is not universally true in the context of set theory.

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.

Subset
In set theory, a subset is a set where every element is also part of another set. If we say that set \(A\) is a subset of set \(B\), it means every element in \(A\) is also contained in \(B\). This relationship is denoted as \(A \subseteq B\).

Subsets can be proper or improper. A proper subset means all the elements of \(A\) are in \(B\), but \(B\) contains additional elements not found in \(A\). An improper subset means \(A\) contains all elements of \(B\), possibly including all elements of \(B\) without any difference.

Understanding subsets is crucial when analyzing situations involving set membership and relationships. It helps identify why conditions may or may not hold true based on how elements are distributed among sets.
Counterexample
A counterexample is a specific example that demonstrates a mathematical statement to be false. To disprove a statement, finding just one valid counterexample suffices. In mathematics, and specifically in set theory, a counterexample can illuminate whether hypothesized relationships between sets hold in all cases.

For instance, in the given exercise, the statement "if \(A \subseteq B\) and \(B subseteq C\), then \(A subseteq C\)" was disproved using the counterexample: \(A = \{1\}\), \(B = \{1, 2\}\), and \(C = \{1, 3\}\). Here, while \(B\) is not a subset of \(C\), \(A\) remains a subset of \(C\). This counterexample invalidates the universal truth of the statement across all possible sets.
Mathematical Statement Disproval
Disproving a mathematical statement involves showing that the statement does not hold under certain conditions. This typically means providing a case where the statement's conclusion is not true, thereby proving it false in a general sense.

In proving statements false, we rely heavily on counterexamples as concrete proof. Continuing with the examples provided in the step-by-step solution, by demonstrating a situation where \(A \subseteq B\) and \(B subseteq C\) do not lead to \(A subseteq C\), we disproved the statement.

Overall, disproving mathematical statements requires critical thinking and validation through specific example scenarios. It emphasizes the need to question and thoroughly test our understanding of mathematical relationships, especially in abstract fields like set theory.

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

A shipment of 24 new cars contains 15 in excellent condition, six with minor defects and three with major defects. If two cars are selected from the shipment, what is the probability that (a) both are in excellent condition? (b) both have minor defects? (c) at most one has a minor defect? (d) at least one has a minor defect? (c) exactly one has a minor defect? (f) neither has a minor defect? How are the results in parts (b), (e), and (f) related?

One quarter of the five-element subsets of \(\\{1,2,3, \ldots, n\\}\) contain the element 7 . Determine \(n(\geq 5)\)

Let \(A=\\{1,2,3, \ldots, 15\\}\) a) How many subsets of \(A\) contain all of the odd integers in \(A\) ? b) How many subsets of \(A\) contain exactly three odd integers? c) How many eight-element subsets of \(A\) contain exactly three odd integers? d) Write a computer program (or develop an algorithm) to generate an eight- element subset of \(A\) and have it print out how many of the eight elements are odd.

Determine whether each of the following statements is true or false. For each false statement, give a counterexample. a) If \(A\) and \(B\) are infinite sets, then \(A \cap B\) is infinite. b) If \(B\) is infinite and \(A \subset B\), then \(A\) is infinite. c) If \(A \subset B\) with \(B\) finite, then \(A\) is finite. d) If \(A \subseteq B\) with \(A\) finite, then \(B\) is finite.

The freshman class of a private engineering college has 300 students. It is known that 180 can program in Pascal, 120 in FORTRAN, 30 in APL, 12 in Pascal and APL, 18 in FORTRAN and APL, 12 in Pascal and FORTRAN, and 6 in all three languages. a) A student is selected at random. What is the probability that she can program in exactly two languages? b) Two students are selected at random. What is the probability that they can i) both program in Pascal? i) both program only in Pascal?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.