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Chapter 5: Problem 21

Write all possible subsets of the given set. $$\\{2\\}$$

Short Answer

Expert verified
The subsets are \(\emptyset\) and \(\{2\}\).

Step by step solution

01

Understanding Subsets

A subset is a set composed of elements from another set. Every element of the subset is also an element of the original set. For any set, the number of possible subsets is given by \(2^n\), where \(n\) is the number of elements in the set.
02

Calculate Number of Subsets

Determine the number of elements in the set. Here, \(\{2\}\) has one element. Thus, the number of possible subsets is \(2^1 = 2\).
03

Identify the Subsets

The subsets of any set include the empty set (denoted as \(\emptyset\)) and the set itself. Therefore, the subsets of \(\{2\}\) are \(\emptyset\) and \(\{2\}\).

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Key Concepts

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

Empty Set
In the realm of set theory, the empty set holds a special place. It's a set that contains no elements at all. In mathematical notation, the empty set is denoted by \( \emptyset \). It can also be symbolized by curly brackets with nothing inside: \( \{\} \). The empty set is considered a subset of every set, including itself. This means that even when a set appears to have no subsets, the empty set is always there, providing at least one subset. This is a fundamental concept in understanding how subsets work.
Number of Subsets Formula
To determine how many subsets a given set has, we use the number of subsets formula. This is given by \(2^n\), where \(n\) represents the number of elements in the original set. For example, for the set \(\{2\}\), which has one element, we calculate the number of subsets as follows:
  • Identify the number of elements: \(n = 1\)
  • Apply the formula: \(2^n = 2^1 = 2\)
Hence, the set \(\{2\}\) has 2 subsets. Understanding this formula is crucial for quickly determining the number of possible subsets without the need to list them all.
Power Set
The power set is a concept within set theory that includes all possible subsets of a given set. This means that for any set, its power set contains:
  • The empty set
  • Each individual element as a subset
  • All possible combinations of the elements
For the set \(\{2\}\), the power set is \( \{\emptyset, \{2\} \} \). Noticing how the power set contains each possible subset can help illuminate how the relationships in data are structured.
Set Theory
Set theory is the mathematical study of sets, which are essentially collections of objects. Sets are fundamental objects in mathematics, and their examination forms the basis for more complex mathematical concepts. In set theory, we describe various operations on sets, such as union, intersection, and complement, but subsets and power sets are the foundational concepts. Understanding the basics, like what sets and subsets are, paves the way towards dealing with these operations effectively. The study of set theory opens a broad avenue of thinking about how things relate to each other in mathematics, coding, databases, and more.

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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?

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

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

Consider the data in Table 5.4 .7 for the amount of oil used annually to generate electricity in the United States from 2000 to 2011 . $$\begin{array}{cc} \hline \begin{array}{c} \text { Years since } \\ 2000 \end{array} & \begin{array}{c} \text { Amount of Oil Used } \\ \text { (millions of gallons) } \end{array} \\ \hline x_{0}=0 & y_{0}=2216 \\ x_{1}=1 & y_{1}=2213.9 \\ x_{2}=2 & y_{2}=2211.8 \\ x_{3}=3 & y_{3}=2209.7 \\ x_{4}=4 & y_{4}=2207.6 \\ x_{5}=5 & y_{5}=2205.5 \\ x_{6}=6 & y_{6}=2203.4 \\ x_{7}=7 & y_{7}=2201.3 \\ x_{8}=8 & y_{8}=2199.2 \\ x_{9}=9 & y_{9}=2197.1 \\ x_{10}=10 & y_{10}=2195 \end{array}$$ Calculate \(\sum_{i=0}^{5} y_{i}\) and interpret.

A contingency table is given without row and column totals. Determine the cardinality in each case. $$\begin{array}{lcc}\hline & \text { Set A } & \text { Set B } \\\\\hline \text { Set C } & 26 & 15 \\\\\text { Set D } & 11 & 28 \\\\\hline\end{array}$$ $$n(B \cup C)$$
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