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Chapter 8: Problem 1

Fill in the blanks. The ___ of two sets is the set of elements that are common to both sets and the ___of two sets is the set of elements that are in one set, or the other, or both.

Short Answer

Expert verified
Intersection; Union.

Step by step solution

01

Understanding Set Theory Terms

In set theory, we need to understand the specific terms used to describe different operations involving sets. Two common operations are the intersection and union of sets.
02

Identify Intersection of Sets

The intersection of two sets is defined as the set of elements that are present in both sets. This means we are looking for elements that are shared between the two sets.
03

Identify Union of Sets

The union of two sets is the set of elements that are present in either of the two sets, or in both. It combines all elements from both sets, without repetition.
04

Filling the Blanks

Using the definitions, we can fill in the blanks: The intersection of two sets is the set of elements that are common to both sets and the union of two sets is the set of elements that are in one set, or the other, or both.

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Key Concepts

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

Intersection of Sets
In set theory, the intersection of sets focuses on the commonalities between them. If you have two sets, say, Set A and Set B, the intersection is made up of elements found in both sets. To symbolize the intersection, we use \( \cap \). If \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \), then the intersection \( A \cap B \) would be \( \{2, 3\} \). These elements 鈥� 2 and 3 鈥� are the ones sharing space in both A and B.

Key points to remember about intersection of sets:
  • Only the common elements are included.
  • If there are no common elements, the intersection results in an empty set, represented by \( \emptyset \).
  • The process is commutative: \( A \cap B = B \cap A \).
Union of Sets
The union of sets gathers all elements across the involved sets. Consider Set A and Set B again. The union, symbolized by \( \cup \), will incorporate every element from both sets. If we continue with our Set A as \( \{1, 2, 3\} \) and Set B as \( \{2, 3, 4\} \), the union \( A \cup B \) becomes \( \{1, 2, 3, 4\} \). Notice how each element is represented only once.

A few essentials regarding union of sets:
  • Includes all elements from the involved sets, without duplication.
  • Zero overlap needed 鈥� even if sets have no common elements, their union is still populated with all distinct entries.
  • Like intersection, union is commutative: \( A \cup B = B \cup A \).
Elements in Sets
Elements in sets are the distinct objects or numbers that form a set. They are typically enclosed in curly braces within set notation. For example, in the set \( \{2, 4, 6\} \), 2, 4, and 6 are the elements. Understanding elements is foundational because they are the building blocks of all set operations and definitions.

To handle elements in sets:
  • Pay attention to uniqueness; each element in a set is distinct 鈥� no repeats.
  • The order of elements does not affect the set. \( \{1,2\} \) is the same as \( \{2,1\} \).
  • Elements can be anything: numbers, letters, or even other sets.

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Most popular questions from this chapter

Breathing Capacity. When fitness instructors prescribe exercise workouts for elderly patients, they must take into account age-related loss of lung function. Studies show that the percent of remaining breathing capacity for someone over 30 years old can be modeled by a linear function. (Source: alsearsmd.com) a. At 35 years of age, approximately \(90 \%\) of maximal breathing capacity remains and at 55 years of age, approximately \(66 \%\) of maximal breathing capacity remains. Let \(a\) be the age of a patient and \(L\) be the percent of her maximal breathing capacity that remains. Write a linear function \(L(a)\) to model this situation. b. Use your answer to part a to estimate the percent of maximal breathing capacity that remains in an 80 -year-old.

Explain why we can think of a function as a machine.

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Find the domain of each function. See Example \(6 .\) a. \(f(x)=x^{2}\) b. \(s(x)=\frac{9}{2 x+1}\)

Perform the operations and simplify, if possible. See Example \(8 .\) $$\frac{2}{5 a}+\frac{3}{2 a}$$
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