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Chapter 2: Problem 37

In Exercises 33-46, determine which sets are the empty set. \(\\{x \mid x\) is the number of women who served as U.S. president before 2016\\}

Short Answer

Expert verified
The set described by \({x \mid x\} is the number of women who served as U.S. president before 2016\)' is an empty set as no woman has served as a U.S. President before 2016, thus meaning no numbers or elements within this set.

Step by step solution

01

Interpret the Set Description

The set includes all numbers \( x \) that represent the number of women who served as U.S. President prior to 2016.
02

Use Factual Knowledge

According to established historical facts, no woman served as President of the United States before 2016.
03

Determine the Set

Since there were no women who served as U.S. Presidents before 2016, there are no elements (no numbers/ \( x \) ) that can belong to the set under this description. Therefore, the set does not contain any elements.
04

Conclusions about the Set

A set that contains no elements is defined as an 'empty set' or 'null set'. Because the given set contains no elements, as determined in the previous steps, it is considered an empty set.

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Key Concepts

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

Empty Set
The concept of an 'empty set' is quite fascinating in set theory. An empty set, often denoted by \( \emptyset \) or \( \{ \} \), is a set that contains no elements. It is important because it serves as a foundational building block in mathematics and logic.

In the context of the exercise given, the set describing women who served as U.S. Presidents before 2016 is empty. Why is that? Because historical records show that no woman held the office before that year, making it impossible for the set to contain any elements. Think of the empty set as the mathematical version of a concept we understand well: having zero of something.
  • The empty set is unique; there is only one empty set up to isomorphism.
  • It's crucial in proving properties about other sets and used commonly in proofs.
  • Mathematicians consider the empty set a subset of every other set, which sometimes helps in establishing more complex mathematical theories.
Set Description
Describing sets in mathematics involves detailing their elements or characteristics. This is often done using set-builder notation, which is seen in the given exercise: \( \{x \mid x \text{ is the number of women who served as U.S. President before 2016} \} \).

In this form of notation, the set is defined by a property that its members, referred to as \( x \) in this case, must satisfy. However, because no such women served as U.S. Presidents before 2016, the described set lacks elements, leading us to identify it as an empty or null set.
  • Set descriptions can be limited by factual or logical constraints, just like in the example from our exercise.
  • Sets are a vital part of different mathematical fields, including algebra, calculus, and discrete mathematics.
  • The ability to describe sets accurately is essential in mathematics as it aids in communication of complex ideas.
Historical Facts
Understanding historical facts is crucial, especially when solving exercises that depend on them. In this exercise, it's notable that no woman had served as U.S. President before 2016. This factual information directly influences our understanding of the set, leading us to correctly determine it as empty.

Historical facts act as a basis for determining the reality of a scenario or exercise. In set theory, they can dictate whether a set has elements or remains empty. It's essential to cross-reference such facts with reliable sources to ensure accuracy.
  • Incorporating historical data in solving mathematical problems helps ground abstract concepts in real-world scenarios.
  • Fact-based problems also illustrate the multidisciplinary nature of mathematical applications.
  • Using historical facts in exercises provides an engaging learning approach, fostering curiosity and enhancing retention.
Null Set
A 'null set' is simply another name for the empty set, denoting a set with zero elements. In mathematics, the terms "null set" and "empty set" are interchangeable, though "empty set" is more commonly used. These concepts form a vital part of set theory and are used extensively across various branches of mathematics.

The null set is as important as sets with elements because it helps define operations like union and intersection and properties like subsets.
  • The null set symbol \( \emptyset \) is used universally, and its representation and concept are understood globally.
  • In operations, the union of any set with an empty set results in the original set, while the intersection with any set results in the empty set.
  • Understanding the null set is foundational for advancing in topics like probability, where empty events may occur.

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Most popular questions from this chapter

A pollster conducting a telephone poll of a city's residents asked two questions: 1\. Do you currently smoke cigarettes? 2\. Regardless of your answer to question 1, would you support a ban on smoking in all city parks? a. Construct a Venn diagram that allows the respondents to the poll to be identified by whether or not they smoke cigarettes and whether or not they support the ban. b. Write the letter b in every region of the diagram that represents smokers polled who support the ban. c. Write the letter c in every region of the diagram that represents nonsmokers polled who support the ban. d. Write the letter d in every region of the diagram that represents nonsmokers polled who do not support the ban.

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

In the August 2005 issue of Consumer Reports, readers suffering from depression reported that alternative treatments were less effective than prescription drugs. Suppose that 550 readers felt better taking prescription drugs, 220 felt better through meditation, and 45 felt better taking St. John's wort. Furthermore, 95 felt better using prescription drugs and meditation, 17 felt better using prescription drugs and St. John's wort, 35 felt better using meditation and St. John's wort, 15 improved using all three treatments, and 150 improved using none of these treatments. (Hypothetical results are partly based on percentages given in Consumer Reports.) a. How many readers suffering from depression were included in the report? Of those included in the report, b. How many felt better using prescription drugs or meditation? c. How many felt better using St. John's wort only? d. How many improved using prescription drugs and meditation, but not St. John's wort? e. How many improved using prescription drugs or St. John's wort, but not meditation? f. How many improved using exactly two of these treatments? g. How many improved using at least one of these treatments?

Describe what is meant by the complement of a set.

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