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Chapter 2: Problem 11

Use Venn diagrams to verify the following two relationships for any events \(A\) and \(B\) (these are called De Morgan's laws): a. \((A \cup B)^{\prime}=A^{\prime} \cap B^{\prime}\) b. \((A \cap B)^{\prime}=A^{\prime} \cup B^{\prime}\)

Short Answer

Expert verified
Verify both De Morgan's laws by ensuring identical shaded regions in each comparison.

Step by step solution

01

Understand De Morgan's Laws

The two De Morgan's laws for sets are: 1. The complement of the union of two sets is the intersection of their complements: \((A \cup B)' = A' \cap B'\).2. The complement of the intersection of two sets is the union of their complements: \((A \cap B)' = A' \cup B'\). Our task is to verify these identities using Venn diagrams.
02

Draw Venn Diagram for \((A \cup B)\)

Draw a rectangle to represent the universal set, inside which we draw two overlapping circles for sets \(A\) and \(B\). Shade the area representing \(A \cup B\), which includes all regions covered by either circle \(A\) or circle \(B\).
03

Find the Complement \((A \cup B)'\)

Determine the regions outside \(A \cup B\). This is all the area outside the shaded region representing \(A \cup B\). Shade this new region.
04

Draw Venn Diagram for \(A' \cap B'\)

In the same universal set, shade the region \(A'\), which is all outside circle \(A\), and \(B'\), which is all outside circle \(B\). The intersection \(A' \cap B'\) is the area where the shaded regions for both \(A'\) and \(B'\) overlap.
05

Compare Results for Part a

Both diagrams for \((A \cup B)'\) and \(A' \cap B'\) should have identical shaded regions, confirming that \((A \cup B)' = A' \cap B'\).
06

Draw Venn Diagram for \((A \cap B)\)

Reuse the Venn diagram setup with the universal set and circles for \(A\) and \(B\). Shade the region \(A \cap B\), which is just the area where circles \(A\) and \(B\) overlap.
07

Find the Complement \((A \cap B)'\)

Determine the regions outside the overlap of \(A\) and \(B\). Shade the areas outside \(A \cap B\) in the Venn diagram.
08

Draw Venn Diagram for \(A' \cup B'\)

In the universal set, shade \(A'\), the area outside \(A\), and \(B'\), the area outside \(B\). \(A' \cup B'\) includes all areas that are shaded in either \(A'\) or \(B'\).
09

Compare Results for Part b

Both diagrams for \((A \cap B)'\) and \(A' \cup B'\) should have identical shaded regions, verifying \((A \cap B)' = A' \cup B'\).

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Key Concepts

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

De Morgan's Laws
De Morgan's Laws are fundamental principles in set theory and logic. They illustrate how the complement of a set operation, like union or intersection, relates to the union or intersection of complements. These laws are named after Augustus De Morgan, a British mathematician. There are two main aspects:

  • **The complement of the union** of two sets is the intersection of their complements:

\[(A \cup B)' = A' \cap B'\]
  • **The complement of the intersection** of two sets is the union of their complements:

\[(A \cap B)' = A' \cup B'\]
Using Venn diagrams can help visually verify these laws by illustrating the areas of overlap and exclusion for various set operations. Students should shade regions step-by-step to accurately see how De Morgan鈥檚 Laws hold true.
Set Theory
Set theory is a branch of mathematical logic that deals with the collection of objects, which we refer to as 'sets'. It forms the foundation of various other areas in mathematics, allowing us to talk about groups of things in a systematic way.

Basic definitions in set theory include:

  • **Sets**: These are collections of distinct elements or members.
  • **Subsets**: A set is a subset of another set if all elements of the first set are also elements of the second.
  • **Universal Set**: This is the set that contains all objects under consideration, typically represented by a rectangle in Venn diagrams.

Understanding these basic concepts is critical in using tools like Venn diagrams to comprehend more complex operations involving sets.
Complement of Sets
The complement of a set is an essential concept in set theory. It refers to all elements not in the given set but contained in the universal set. If set \(A\) is a subset of some universal set \(U\), the complement of \(A\) (denoted as \(A'\)) is the collection of elements in \(U\) that are not in \(A\).

To find the complement:

  • Identify the universal set \(U\).
  • Recognize the elements in the subset \(A\).
  • The complement \(A'\) consists of everything in \(U\) but not in \(A\).

Venn diagrams visually demonstrate this by shading different areas for sets and their complements, making these abstract concepts more intuitive.
Union and Intersection of Sets
In set theory, union and intersection are two fundamental operations used to combine and relate sets. Understanding these concepts is crucial for solving various set-based problems.

The **union** of two sets \(A\) and \(B\), denoted \(A \cup B\), is a set containing all elements that are in \(A\), in \(B\), or in both. Imagine it as the combination of everything in both sets.

The **intersection** of two sets \(A\) and \(B\), denoted \(A \cap B\), includes only the elements present in both \(A\) and \(B\). It's where the sets overlap.

Using Venn diagrams:

  • Shade both circles for union \(A \cup B\).
  • Shade only the overlapping region for intersection \(A \cap B\).

Visualizing these operations with Venn diagrams helps to clarify how sets relate to one another in different operations.

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Most popular questions from this chapter

Fasteners used in aircraft manufacturing are slightly crimped so that they lock enough to avoid loosening during vibration. Suppose that \(95 \%\) of all fasteners pass an initial inspection. Of the \(5 \%\) that fail, \(20 \%\) are so seriously defective that they must be scrapped. The remaining fasteners are sent to a recrimping operation, where \(40 \%\) cannot be salvaged and are discarded. The other \(60 \%\) of these fasteners are corrected by the recrimping process and subsequently pass inspection. a. What is the probability that a randomly selected incoming fastener will pass inspection either initially or after recrimping? b. Given that a fastener passed inspection, what is the probability that it passed the initial inspection and did not need recrimping?

Show that \(\left(\begin{array}{c}n \\\ k\end{array}\right)=\left(\begin{array}{c}n \\ n-k\end{array}\right)\). Give an interpretation involving subsets.

The three major options on a car model are an automatic transmission \((A)\), a sunroof \((B)\), and an upgraded stereo \((C)\). If \(70 \%\) of all purchasers request \(A, 80 \%\) request \(B, 75 \%\) request \(C, 85 \%\) request \(A\) or \(B, 90 \%\) request \(A\) or \(C, 95 \%\) request \(B\) or \(C\), and \(98 \%\) request \(A\) or \(B\) or \(C\), compute the probabilities of the following events. [Hint: "A or \(B^{\prime \prime}\) is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.] a. The next purchaser will request at least one of the three options. b. The next purchaser will select none of the three options. c. The next purchaser will request only an automatic transmission and neither of the other two options. d. The next purchaser will select exactly one of these three options.

A personnel manager is to interview four candidates for a job. These are ranked \(1,2,3\), and 4 in order of preference and will be interviewed in random order. However, at the conclusion of each interview, the manager will know only how the current candidate compares to those previously interviewed. For example, the interview order \(3,4,1,2\) generates no information after the first interview, shows that the second candidate is worse than the first, and that the third is better than the first two. However, the order 3,4 , 2,1 would generate the same information after each of the first three interviews. The manager wants to hire the best candidate but must make an irrevocable hire/no hire decision after each interview. Consider the following strategy: Automatically reject the first \(s\) candidates and then hire the first subsequent candidate who is best among those already interviewed (if no such candidate appears, the last one interviewed is hired). For example, with \(s=2\), the order \(3,4,1\), 2 would result in the best being hired, whereas the order \(3,1,2,4\) would not. Of the four possible \(s\) values \((0,1,2\), and 3\()\), which one maximizes \(P\) (best is hired)? [Hint: Write out the 24 equally likely interview orderings: \(s=0\) means that the first candidate is automatically hired.]

An aircraft seam requires 25 rivets. The seam will have to be reworked if any of these rivets is defective. Suppose rivets are defective independently of one another, each with the same probability. a. If \(20 \%\) of all seams need reworking, what is the probability that a rivet is defective? b. How small should the probability of a defective rivet be to ensure that only \(10 \%\) of all seams need reworking?
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