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Chapter 2: Problem 99

Let $$ \begin{aligned} U &=\\{x \mid x \in \mathbf{N} \text { and } x<9\\} \\ A &=\\{x \mid x \text { is an odd natural number and } x<9\\} \\ B &=\\{x \mid x \text { is an even natural number and } x<9\\} \\ C &=\\{x \mid x \in \mathbf{N} \text { and } 1

Short Answer

Expert verified
The intersection of the sets \(A \cap U\) is {1, 3, 5, 7}.

Step by step solution

01

Identify the elements of set A

The set A is defined as the set of all odd natural numbers less than 9. Therefore, A = {1, 3, 5, 7}.
02

Identify the elements of set U

The set U is defined as the set of all natural numbers less than 9. So, U = {1, 2, 3, 4, 5, 6, 7, 8}.
03

Find the intersection of A and U

The intersection of the sets A and U, \(A \cap U\) , represents all the elements that are common to both A and U. So, \(A \cap U\) = {1, 3, 5, 7}. This can be obtained by finding the common elements in both sets.

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Key Concepts

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

Natural Numbers
The concept of natural numbers is foundational in mathematics. Natural numbers, often symbolized by the letter \( \mathbf{N} \), are the basic building blocks of arithmetic. These are positive integers that start from 1 and go onwards.
  • Natural numbers do not include zero, fractions, or negative numbers.
  • Examples of natural numbers are \( 1, 2, 3, 4, \text{and so on}. \)
  • These numbers are used for counting and ordering.
In many exercises, natural numbers are involved in defining sets and performing operations like intersections and unions.
Odd Numbers
Odd numbers are a specific subset of natural numbers. They are integers that cannot be evenly divided by 2. Understanding odd numbers helps in solving problems that require distinguishing between different types of integers.
  • Examples of odd numbers include \( 1, 3, 5, 7, \text{etc.} \)
  • When an odd number is divided by 2, the remainder is always 1.
  • In set theory problems, sets often specifically define using odd numbers for particular conditions or criteria.
The exercise you encountered involved finding odd natural numbers less than 9, which gave us the set \( A = \{1, 3, 5, 7\} \).
Intersection of Sets
The intersection of two sets is a fundamental operation in set theory. It provides a way to find common elements between sets, an important technique for analyzing relationships and solving problems. The intersection of sets \( A \) and \( B \), written as \( A \cap B \), includes all elements that both sets share.
  • If \( A = \{1, 3, 5, 7\} \) and \( U = \{1, 2, 3, 4, 5, 6, 7, 8\} \), then \( A \cap U = \{1, 3, 5, 7\} \).
  • The process requires checking each element of \( A \) to see if it is also in \( U \).
  • Only elements present in both sets appear in the resulting intersection.
This operation is essential for solving many mathematical problems and is widely used in computer science and various fields of research.

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Most popular questions from this chapter

In the August 2005 issue of Conswner 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?

In Exercises 41-66, 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. \(A \cup C^{\prime}\)

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

Use the formula for the cardinal number of the union of two sets to solve Exercises 93-96. Set \(A\) contains 8 letters and 9 numbers. Set \(B\) contains 7 letters and 10 numbers. Four letters and 3 numbers are common to both sets \(A\) and \(B\). Find the number of elements in set \(A\) or set \(B\).

Assume \(A \neq B\). Draw a Venn diagram that correctly illustrates the relationship between the sets. \(A \cup B=A\)
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