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Chapter 5: Problem 14

Let \(U=\\{x | x \text { is a whole number and } 1 \leq x \leq 15\\}\) \(A=\\{1,2,3,4,5\\}, B=\\{2,4,6,8,10\\},\) and \(C=\\{11,12,13,14,15\\} .\) Determine the cardinality of the indicated sets. $$n(C)$$

Short Answer

Expert verified
The cardinality of set C is 5.

Step by step solution

01

Understand the Problem

We need to find the cardinality of set \(C\), which means we want to know the number of elements in set \(C\). The set \(C\) is defined as \(C=\{11,12,13,14,15\}\).
02

List the Elements of Set C

Write down all the elements of set \(C\). The elements are \(11, 12, 13, 14,\) and \(15\).
03

Count the Elements in Set C

Count the number of elements listed in set \(C\).
04

Result of the Cardinality

The number of elements in set \(C\) is 5. Therefore, \(n(C) = 5\).

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Key Concepts

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

Whole Numbers
Whole numbers are a fundamental part of mathematics and include all the natural numbers such as 0, 1, 2, 3, and so on. Essentially, whole numbers are numbers that have no fractional or decimal component.
It's important to remember that while natural numbers start from 1, whole numbers start from 0.

In the exercise, the universal set \( U \) containing whole numbers is defined from 1 to 15. This means \( U \) consists exclusively of the whole numbers: 1, 2, 3 up to 15.
  • Whole numbers can be positive but never negative.
  • They are used in counting and ordering.
  • Whole numbers form the basis for other number sets like integers and real numbers.
Understanding whole numbers helps us work efficiently with various mathematical concepts like set theory, especially when identifying or counting elements within a set.
Set Theory
Set theory is a branch of mathematics that studies collections of objects, known as sets. A set is defined by its distinct elements.
In our scenario, sets \( A \), \( B \), and \( C \) contain different elements as specified:
  • \( A = \{1,2,3,4,5\} \)
  • \( B = \{2,4,6,8,10\} \)
  • \( C = \{11,12,13,14,15\} \)
Set theory provides the tools to describe the relationships and operations like unions and intersections between these sets.

Key terms in set theory:
  • Universal Set \( U \): The complete set containing all possible elements under consideration, in this case, from 1 to 15.
  • Cardinality: The number of elements in a set. In this exercise, we found that the cardinality \( n(C) = 5 \) for set \( C \).
  • Subset: A set whose elements are all contained in another set.
By understanding set theory, students can solve more complex problems and establish relationships between different sets.
Elementary Math
Elementary mathematics forms the foundation of advanced mathematical concepts. It includes basic number operations, introduction to sets, geometry, and more.
In the context of this exercise, focusing on identifying and counting elements in a set is a part of elementary math, crucial for developing logical thinking and problem-solving skills in students.
  • Elementary math simplifies complex problems by breaking them down into basic concepts.
  • It encourages the use of logical reasoning to find solutions.
  • Developing these skills in early stages forms a solid base for higher-level math and real-world applications.
By practicing problems such as counting elements in a set and determining cardinality, students enhance their basic math ability, preparing them for more challenging mathematical explorations.

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Most popular questions from this chapter

Consider a March 2015 Gallup poll in which 1025 randomly chosen American adults were asked if they believe that more emphasis should be placed on traditional sources of energy, such as fossil fuels and nuclear energy, or on alternative energy sources, such as wind and solar energy. The respondents were also asked to classify their party affiliation. The results are shown in the contingency table. $$\begin{array}{lccc}\hline & \begin{array}{c}\text { More Traditional } \\\\\text { Sources }\end{array} & \begin{array}{c}\text { More Alternative } \\\\\text { Sources }\end{array} & \text { Total } \\\\\hline \text { Democrat } & 107 & 498 & 605 \\\\\text { Republican } & 252 & 168 & 420 \\ \text { Total } & 359 & 666 & 1025 \\\\\hline\end{array}$$ Answer the following questions. Round to the nearest whole percentage as necessary. What percentage of those in the survey said that more emphasis should be placed on alternative energy sources and are Democrats?

Consider the set of bivariate data shown in the table, and compute the indicated sums. $$\begin{array}{cccccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ y & 5 & 16.5 & 20 & 21.2 & 21.8 & 22.1 & 22.4 \\ \hline \end{array}$$ $$(\Sigma x)^{2}-\Sigma x^{2}$$

Combine the Multiplication Principle and combinations to answer the questions.In a shipment of 30 portable stereos, 8 are known to be defective. In how many ways can a sample of 10 be chosen so that 2 are defective and 8 are not defective?

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. $$(A \cup U)^{C}$$

Calculate the given permutation. Express large values using Enotation with the mantissa rounded to two decimal places.$$_{9} P_{3}$$
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