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Chapter 5: Problem 33

Let \(U=\\{x | x \text { is a natural number and } 1 \leq x \leq 12\\}\) \(A=\\{1,2,3,4,5\\}, B=\\{2,4,6,8,10\\}\) and \(C=\\{3,6,9,12\\} .\) Determine the cardinality of the indicated sets. $$B^{C}$$

Short Answer

Expert verified
The cardinality of \( B^c \) is 7.

Step by step solution

01

Determine the Universal Set

The universal set \( U \) is given by \( U = \{ x | x \text{ is a natural number and } 1 \leq x \leq 12 \} \). This means \( U = \{ 1, 2, 3, ..., 12 \} \).
02

Identify Elements in Set B

The set \( B = \{ 2, 4, 6, 8, 10 \} \) includes the even natural numbers between 1 and 12.
03

Complement of Set B in U

\( B^c \) refers to the complement of set \( B \) with respect to the universal set \( U \). This means we list all elements in \( U \) that are not in \( B \).
04

Calculate B^C

Since \( B = \{ 2, 4, 6, 8, 10 \} \), then \( B^c = \{ 1, 3, 5, 7, 9, 11, 12 \} \), which are all elements in \( U \) not present in \( B \).
05

Determine the Cardinality of B^C

Count the elements in \( B^c \). We have \( B^c = \{ 1, 3, 5, 7, 9, 11, 12 \} \), which contains 7 elements. Thus, the cardinality of \( B^c \) is 7.

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Key Concepts

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

Universal Set
The concept of a universal set is fundamental in set theory. A universal set contains all possible elements under consideration for a particular discussion or problem. In our exercise, the universal set \( U \) is defined as all natural numbers between 1 and 12:
  • This means \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\} \).
A universal set serves as the 'universe' from which sets can be derived, and it provides the complete context for discussing any subsets within a particular range or criteria. Understanding the universal set is crucial because it helps to identify which elements are included and which are not when exploring the relationships between different sets.
Set Complement
The complement of a set is an important idea in set theory, denoted as \( B^c \) for the complement of \( B \). The complement includes all elements that are in the universal set \( U \), but not in the subset \( B \).
In our exercise, the set \( B \) is defined as the even numbers within our universal set:
  • \( B = \{ 2, 4, 6, 8, 10 \} \)
To find the complement \( B^c \), we subtract these elements from the universal set \( U \). This gives us the numbers in \( U \) that are not in \( B \):
  • \( B^c = \{ 1, 3, 5, 7, 9, 11, 12 \} \)
Understanding set complements helps you identify what is "outside" a certain set when viewed in the context of the universal set.
Cardinality
Cardinality in set theory refers to the count of elements in a set. It gives a measure of the "size" of a set. For any given set \( S \), its cardinality is denoted as \( |S| \). In our exercise, we calculated the cardinality of the set complement \( B^c \).
  • First, note that \( B^c = \{ 1, 3, 5, 7, 9, 11, 12 \} \).
  • The cardinality of \( B^c \) is the number of elements in this complement set.
  • By counting these elements, we determine that \( |B^c| = 7 \).
Understanding the cardinality is important as it provides a quantitative understanding of a set's elements and allows you to compare different sets effectively.
Natural Numbers
Natural numbers are the set of positive integers that begin from 1 and continue incrementally:
  • \( \{ 1, 2, 3, 4, 5, \ldots \} \)
Natural numbers are foundational in mathematics and are used to count and order objects. In our exercise, the universal set \( U \) consists of natural numbers from 1 to 12. These numbers are the building blocks for the subsets discussed in set theory problems, like the ones identified in the sets \( A, B, \) and \( C \). Understanding natural numbers helps you grasp what elements are being considered when sets are defined or when operations like complements are performed. They offer a clear view of discrete, countable items.

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Most popular questions from this chapter

Let \(U=\\{x | x \text { is a natural number and } 1 \leq x \leq 12\\}\) \(A=\\{1,2,3,4,5\\}, B=\\{2,4,6,8,10\\}\) and \(C=\\{3,6,9,12\\} .\) Determine the cardinality of the indicated sets. $$(B \cup C)^{C}$$

How many different ways can the letters of the word DESIGN be arranged?

Use the given \(n=5\) ordered pairs to calculate the indicated sums. Round to two decimal places. $$\begin{array}{cccccc} \hline x & 3.15 & 16.83 & 14.15 & 28.81 & 18.20 \\ y & 0.05 & 1.56 & 1.98 & 2.17 & 1.77 \\ \hline \end{array}$$ $$\sqrt{\Sigma x y}$$

Use the given \(n=5\) ordered pairs to calculate the indicated sums. Round to two decimal places. $$\begin{array}{cccccc} \hline x & 3.15 & 16.83 & 14.15 & 28.81 & 18.20 \\ y & 0.05 & 1.56 & 1.98 & 2.17 & 1.77 \\ \hline \end{array}$$ $$\Sigma x y$$

Consider the data in Table 5.4 .8 for the number of confirmed cases of Lyme disease in Vermont annually from 2005 to 2014 . $$\begin{array}{cc} \hline \text { Years since 2000 } & \begin{array}{c} \text { Cases of Lyme Disease } \\ \text { in Vermont } \end{array} \\ \hline x_{5}=5 & y_{5}=54 \\ x_{6}=6 & y_{6}=105 \\ x_{7}=7 & y_{7}=138 \\ x_{8}=8 & y_{8}=330 \\ x_{9}=9 & y_{9}=323 \\ x_{10}=10 & y_{10}=271 \\ x_{11}=11 & y_{11}=476 \\ x_{12}=12 & y_{12}=386 \\ x_{13}=13 & y_{13}=674 \\ x_{14}=14 & y_{14}=599 \\ \hline \end{array}$$ Determine the sum \(\sum_{i=10}^{14} y_{i}\) and interpret.
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