/*! 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 66 \(B \cap(A \cup C)=(A \cap B) \c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 66

\(B \cap(A \cup C)=(A \cap B) \cup(B \cap C)\)

Short Answer

Expert verified
According to the distributive law of sets, the provided solution set is correct.

Step by step solution

01

Identify the given sets

From the given equation: \(B \cap(A \cup C)=(A \cap B) \cup(B \cap C)\), the sets \(A, B, C\) are the given sets.
02

Understand distributive law on sets

The Distributive Law states that the intersection of a set across the union of two other sets equals the union of the intersections of the first set with each of the other two sets. This can be expressed as: \(B \cap(A \cup C)=(B \cap A) \cup(B \cap C)\). So, the problem is correct according this law.
03

Verification of the equality with a Venn Diagram

To visualize and fully validate this equality, one can sketch a Venn diagram, marking the intersections and unions of the sets as necessary and verify whether they satisfy the given equation.

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.

Distributive Law in Sets
The Distributive Law in Set Theory helps us understand how operations like intersection and union interact with each other. It is similar to the distributive law in arithmetic, where multiplication distributes over addition. In sets, we use intersection (\( \cap \)) and union (\( \cup \)) operations instead:
  • The law states that the intersection of one set with the union of two other sets is the same as the union of the intersections of the set with each of the other two sets.
  • Mathematically, it is expressed as \( B \cap (A \cup C) = (B \cap A) \cup (B \cap C) \).
  • This formula helps to simplify set expressions and solve problems related to overlapping groups.

This law is particularly helpful when dealing with complex set operations, breaking them down into simpler components. Understanding this principle can greatly assist in working with set equations effectively.
Venn Diagram
Venn Diagrams are a visual tool used to represent relationships between different sets. They are especially useful in illustrating how different set operations work:
  • A Venn Diagram consists of simple closed figures like circles, each representing a set.
  • Overlaps of these shapes illustrate intersections, while the union covers all the area contained within at least one of the circles.
  • In our problem, drawing a Venn Diagram can help to verify the equation \( B \cap (A \cup C) = (A \cap B) \cup (B \cap C) \).

By sketching the sets and their corresponding intersections and unions as per the equation, you can visually confirm the validity. This makes abstract concepts more tangible and easier to understand.
Intersection and Union of Sets
Intersection and Union are two fundamental concepts in Set Theory. They describe different ways sets can combine or relate:
  • Intersection (\( \cap \)): This operation yields a new set containing all elements common to both sets. For example, \( A \cap B \) includes only elements present in both A and B.
  • Union (\( \cup \)): This operation results in a set containing every element from both sets, without duplication. The union of \( A \) and \( B \), noted as \( A \cup B \), encompasses all unique elements from A and B.
  • These operations are the building blocks of more complex set equations, such as the one in our exercise.

When dealing with multiple sets, recognizing how and when to use intersection and union is crucial. Understanding these concepts helps solve problems involving multiple overlapping groups, making your approach to set problems more comprehensive.

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

Find each of the following sets. \(A \cap \varnothing\)

In Exercises \(1-4\), describe a universal set \(U\) that includes all elements in the given sets. Answers may vary. \(A=\\{\) Pepsi, Sprite \(\\}\) \(B=\\{\) Coca-Cola, Seven-Up \(\\}\)

a. Let \(A=\\{3\\}, B=\\{1,2\\}, C=\\{2,4\\}\), and \(U=\\{1,2,3,4,5,6\\}\). Find \((A \cup B)^{\prime} \cap C\) and \(A^{\prime} \cap\left(B^{\prime} \cap C\right)\). b. Let \(A=\\{\mathrm{d}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\}, B=\\{\mathrm{a}, \mathrm{c}, \mathrm{f}, \mathrm{h}\\}, C=\\{\mathrm{c}, \mathrm{e}, \mathrm{g}, \mathrm{h}\\}\), and \(U=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \ldots, \mathrm{h}\\}\). Find \((A \cup B)^{\prime} \cap C\) and \(A^{\prime} \cap\left(B^{\prime} \cap C\right)\) c. Based on your results in parts (a) and (b), use inductive reasoning to write a conjecture that relates \((A \cup B)^{\prime} \cap C\) and \(A^{\prime} \cap\left(B^{\prime} \cap C\right)\). d. Use deductive reasoning to determine whether your conjecture in part (c) is a theorem.

What is the blood type of a universal recipient?

The group should define three sets, each of which categorizes \(U\), the set of students in the group, in different ways. Examples include the set of students with blonde hair, the set of students no more than 23 years old, and the set of students whose major is undecided. Once you have defined the sets, construct a Venn diagram with three intersecting sets and eight regions. Each student should determine to which region he or she belongs. Illustrate the sets by writing each first name in the appropriate region.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics 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.