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Chapter 2: Problem 97

In Exercises 97-104, 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 union of set A and B, \(A 鈭� B\), is \{1, 2, 3, 4, 5, 6, 7, 8\}

Step by step solution

01

Identifying elements in set A

From the problem, set A is defined as all odd natural numbers less than 9. Therefore, set A is \{1, 3, 5, 7\}.
02

Identifying elements in set B

Similarly, set B is defined as all even natural numbers less than 9. Hence, set B is \{2, 4, 6, 8\}.
03

Finding the Union of Set A and B

The union of set A and B, denoted as A 鈭� B, is the set containing all the elements of set A and set B. To find the union of set A and B, you have to list the elements that exist in either set A, set B, or in both. Each element should be recorded only once. Hence, \(A 鈭� B\) is \{1, 2, 3, 4, 5, 6, 7, 8\}.

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Key Concepts

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

Union of Sets
In set theory, one of the fundamental operations is the union of sets. The union of two sets includes every unique element that appears in either set. If you're given set A and set B, their union, denoted as \(A \cup B\), consists of all elements from both sets without repetition.
To find the union:
  • List all elements from set A.
  • List all elements from set B.
  • Combine these lists, ensuring no duplicates, to create a single set.
For sets A \(\{1, 3, 5, 7\}\) and B \(\{2, 4, 6, 8\}\), the union A 鈭� B results in \(\{1, 2, 3, 4, 5, 6, 7, 8\}\), as it encompasses each number without repetition. Understanding the union operation is crucial, as it's often used to combine data or solve problems involving multiple groups.
Natural Numbers
Natural numbers are the set of positive integers starting from 1 and increasing sequentially by 1 each time, such as 1, 2, 3, and so on. These numbers are used to count objects and measure things that cannot be divided into smaller parts without losing their nature.
Some key characteristics of natural numbers include:
  • They are always whole numbers (no fractions or decimals).
  • They include all positive integers greater than zero.
  • The sequence of natural numbers is infinite.
By understanding natural numbers, we lay the groundwork for more advanced mathematical concepts, such as the classifications into odd and even numbers.
Odd Numbers
Odd numbers are a subset of natural numbers. They cannot be evenly divided by 2, meaning they always leave a remainder of 1 when divided by 2. Odd numbers follow an alternating pattern with even numbers.
Consider these common traits of odd numbers:
  • They are of the form \(n = 2k + 1\), where \(k\) is an integer.
  • The sequence starts with 1 and continues as 1, 3, 5, 7, etc.
  • They are not divisible by 2.
Odd numbers play an important role in various applications, including number theory and patterns within sequences. Recognizing them is essential for solving numerous mathematical problems.
Even Numbers
Even numbers are another group within natural numbers distinguished by their divisibility by 2. This means an even number can be divided into two equal parts without any remainder.
Key features of even numbers include:
  • They take the form \(n = 2k\), where \(k\) is an integer.
  • They begin from 2 and continue as 2, 4, 6, 8, etc.
  • They include any number divisible by 2 with no remainder.
Recognizing even numbers is vital, since they often appear in patterns and calculations across mathematics, making various problem-solving methods more efficient.

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Most popular questions from this chapter

An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 894 drank alcohol regularly, 665 smoked cigarettes, 192 used illegal drugs, 424 drank alcohol regularly and smoked cigarettes, 114 drank alcohol regularly and used illegal drugs, 119 smoked cigarettes and used illegal drugs, 97 engaged in all three behaviors, and 309 engaged in none of these behaviors. Source: Jamie Langille, University of Nevada Las Vegas a. How many students were surveyed? Of those surveyed, b. How many drank alcohol regularly or smoked cigarettes? c. How many used illegal drugs only? d. How many drank alcohol regularly and smoked cigarettes, but did not use illegal drugs? e. How many drank alcohol regularly or used illegal drugs, but did not smoke cigarettes? f. How many engaged in exactly two of these behaviors? g. How many engaged in at least one of these behaviors?

An anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 894 drank alcohol regularly, 665 smoked cigarettes, 192 used illegal drugs, 424 drank alcohol regularly and smoked cigarettes, 114 drank alcohol regularly and used illegal drugs, 119 smoked cigarettes and used illegal drugs, 97 engaged in all three behaviors, and 309 engaged in none of these behaviors. Source: Jamie Langille, University of Nevada Las Vegas a. How many students were surveyed? Of those surveyed, b. How many drank alcohol regularly or smoked cigarettes? c. How many used illegal drugs only? d. How many drank alcohol regularly and smoked cigarettes, but did not use illegal drugs? e. How many drank alcohol regularly or used illegal drugs, but did not smoke cigarettes? f. How many engaged in exactly two of these behaviors? g. How many engaged in at least one of these behaviors?

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

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

Can an \(\mathrm{A}^{+}\)person donate blood to an \(\mathrm{A}^{-}\)person?
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