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Chapter 2: Problem 106

\(\left(A^{\prime} \cap B\right) \cup(A \cap B)\)

Short Answer

Expert verified
The resulting set of the expression \(\left(A^{\prime} \cap B\right) \cup(A \cap B)\) is equal to the set B.

Step by step solution

01

Understanding the Symbols

Firstly, it's important to understand what these symbols mean. In set theory, \(\cap\) stands for intersection, \(\cup\) stands for union, and ' (prime) stands for complement. Intersection of two sets A and B (\(A \cap B\)) includes all the elements which are common to both A and B. Union of two sets A and B (\(A \cup B\)) includes all the elements of A and B. Complement of set A (\(A^{\prime}\)) includes all the elements which are not in A.
02

Break Down the Expression

The given expression is \(\left(A^{\prime} \cap B\right) \cup(A \cap B)\). This can be seen as union (\(\cup\)) of two sub-expressions: \(A^{\prime} \cap B\) and \(A \cap B\). The first sub-expression, \(A^{\prime} \cap B\), means the elements that are in B but not in A. The second sub-expression, \(A \cap B\), means the elements that are common in A and B.
03

Combine the Results

Now, we combine the results from step 2. The result of the union (\(\cup\)) of these two set expressions, \(\left(A^{\prime} \cap B\right)\) and \((A \cap B)\), gives us all the elements that are in B. This is because we've accounted for all elements in B that are not in A and all elements in B that are also in A. So the result of the given expression simplifies to B itself.

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Key Concepts

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

Set Operations
Set operations are fundamental tools in set theory. They allow us to manipulate sets to uncover various relationships between them.
These operations help us understand how sets interact and overlap. The main set operations include:
  • Union (\( \cup \)): Combines all elements from two sets.
  • Intersection (\( \cap \)): Identifies elements common to two sets.
  • Complement (\( A' \)): Includes elements not in a given set.
Understanding these operations is crucial, especially when dealing with more complex expressions. Each operation can be visualized using a Venn diagram, which can help in understanding these concepts better.
Mastery of set operations is key in various fields such as mathematics, computer science, and probability.
Intersection
The intersection of sets is a key concept when we want to find commonalities. When you intersect two sets, you are looking for elements that exist in both sets.
The symbol for intersection is \( \cap \). For example, let's consider sets \( A \) and \( B \). Their intersection \( A \cap B \) contains only the elements that are found in both set \( A \) and set \( B \).
  • If \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), then \( A \cap B = \{2, 3\} \).
  • Visually, in a Venn diagram, the intersection is the overlapping area of the two sets.
Understanding intersections can help solve problems where finding shared elements is essential, such as finding common friends or shared features between two groups.
Union
Union is about combining all elements from two sets into one larger set. The symbol used for union is \( \cup \). When we find the union of two sets \( A \) and \( B \), we are merging these sets together to form a new set.
  • For instance, if \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), then their union is \( A \cup B = \{1, 2, 3, 4, 5\} \).
  • The union combines all elements, but of course, duplicates are omitted, as each element is unique in a set.
The union operation is visually represented in a Venn diagram by shading all parts of the involved circles.
It's a crucial operation because it allows for the combination of datasets for further analysis and insights. For example, combining resources or data from multiple sources.
Complement
The complement of a set focuses on what is outside the set. When we talk about the complement, we're referring to items that are not included in the set under consideration.
In set notation, the complement of set \( A \) is represented as \( A' \).
  • Consider a universal set, \( U = \{1, 2, 3, 4, 5\} \), and a set \( A = \{1, 2\} \).
  • The complement \( A' = \{3, 4, 5\} \) includes all elements of the universal set \( U \) that are not in \( A \).
In Venn diagrams, the complement of a set is shown by shading the area that does not belong to the set within the universal set's boundary.
The concept of complement is widely used in various areas such as logic, probability, and computer science, highlighting what is excluded from a given condition or set.

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Most popular questions from this chapter

I used a Venn diagram to prove that \((A \cup B)^{\prime}\) and \(A^{\prime} \cup B^{\prime}\) are not equal.

Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. Set \(A\) contains 17 elements, set \(B\) contains 20 elements, and 6 elements are common to sets \(A\) and \(B\). How many elements are in \(A \cup B\) ?

If \(A \cap B=\varnothing\), then \(n(A \cup B)=n(A)+n(B)\).

A pollster conducting a telephone poll asked three questions: 1\. Are you religious? 2\. Have you spent time with a person during his or her last days of a terminal illness? 3\. Should assisted suicide be an option for terminally ill people? a. Construct a Venn diagram with three circles that can assist the pollster in tabulating the responses to the three questions. b. Write the letter b in every region of the diagram that represents all religious persons polled who are not in favor of assisted suicide for the terminally ill. c. Write the letter c in every region of the diagram that represents the people polled who do not consider themselves religious, who have not spent time with a terminally ill person during his or her last days, and who are in favor of assisted suicide for the terminally ill. d. Write the letter \(\mathrm{d}\) in every region of the diagram that represents the people polled who consider themselves religious, who have not spent time with a terminally ill person during his or her last days, and who are not in favor of assisted suicide for the terminally ill. e. Write the letter e in a region of the Venn diagram other than those in parts (b)-(d) and then describe who in the poll is represented by this region.

In Exercises 13-24, let $$ \begin{aligned} U &=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}\\} \\\ A &=\\{\mathrm{a}, \mathrm{g}, \mathrm{h}\\} \\\B &=\\{\mathrm{b}, \mathrm{g}, \mathrm{h}\\} \\ C &=\\{\mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}, \mathrm{f}\\} \end{aligned} $$ Find each of the following sets. \((A \cap B \cap C)^{\prime}\)
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