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Chapter 2: Problem 58

List all the subsets of the given set. \(\\{\mathrm{I}, \mathrm{II}, \mathrm{III}\\}\)

Short Answer

Expert verified
The subsets of the set \(\{\mathrm{I}, \mathrm{II}, \mathrm{III}\}\) are: \(\emptyset\), \(\{\mathrm{I}\}\), \(\{\mathrm{II}\}\), \(\{\mathrm{III}\}\), \(\{\mathrm{I}, \mathrm{II}\}\), \(\{\mathrm{I}, \mathrm{III}\}\), \(\{\mathrm{II}, \mathrm{III}\}\), \(\{\mathrm{I}, \mathrm{II}, \mathrm{III}\}\).

Step by step solution

01

List all subsets of length 0

There is always one subset that contains no elements, which is called the empty set. Represented as \(\emptyset\) or \(\{\}\).
02

List all subsets of length 1

The subsets with only one element, also known as the singleton sets, are \(\{\mathrm{I}\}\), \(\{\mathrm{II}\}\), and \(\{\mathrm{III}\}\).
03

List all subsets of length 2

Now list all subsets having two elements. They are \(\{\mathrm{I}, \mathrm{II}\}\), \(\{\mathrm{I}, \mathrm{III}\}\), and \(\{\mathrm{II}, \mathrm{III}\}\).
04

List the subset of length 3

There is only one subset including all the elements from the given set: \(\{\mathrm{I}, \mathrm{II}, \mathrm{III}\}\).

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Key Concepts

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

Subsets
A subset is essentially a collection of elements taken from another set. Suppose we have a set \( A \), a subset of \( A \) contains zero or more elements from \( A \).
  • Every set is a subset of itself.
  • The empty set, which has no elements, is a subset of every set.
Consider the set \( \{\mathrm{I}, \mathrm{II}, \mathrm{III}\} \). Its subsets range from having none of its elements (the empty set) to all its elements. In the solution to the exercise, subsets were listed by their element count:
  • Length 0: \( \emptyset \)
  • Length 1: \( \{\mathrm{I}\}, \{\mathrm{II}\}, \{\mathrm{III}\} \)
  • Length 2: \( \{\mathrm{I}, \mathrm{II}\}, \{\mathrm{I}, \mathrm{III}\}, \{\mathrm{II}, \mathrm{III}\} \)
  • Length 3: \( \{\mathrm{I}, \mathrm{II}, \mathrm{III}\} \)
In total, there are \( 2^n \) subsets, where \( n \) is the number of elements in the original set. For our set, \( n = 3 \), so there are \( 2^3 = 8 \) subsets.
Empty Set
The empty set, also known as the null set, contains no elements. It is denoted by \( \emptyset \) or \( \{\} \).
  • Despite having no elements, it's important as it represents the idea of nothingness within set theory.
  • It is considered a subset of every possible set.
Including the empty set as a subset puts emphasis on the inclusive nature of subsets, ensuring every possible combination is considered. Understanding the empty set helps grasp the foundational elements of set theory and sets the stage for more complex operations and concepts.
Singleton Sets
Singleton sets are subsets containing exactly one element. They are sometimes called single-element subsets.
  • From a given set \( \{\mathrm{I}, \mathrm{II}, \mathrm{III}\} \), the singleton sets are \( \{\mathrm{I}\} \), \( \{\mathrm{II}\} \), and \( \{\mathrm{III}\} \).
  • They emphasize the importance of each individual element within a set.
Even though they appear simple, singleton sets are crucial when examining individual elements and their roles in forming more complex sets. This appreciation for individual elements is pivotal in relation to broader mathematical analysis and operations.
Powerset
The powerset of a set is the set of all its subsets, including the empty set and the set itself. Notationally, if \( A \) is a set, the powerset of \( A \) is often denoted \( \mathcal{P}(A) \).
  • For the set \( \{\mathrm{I}, \mathrm{II}, \mathrm{III}\} \), the powerset contains all of its 8 subsets.
  • The total count of elements in a powerset is \( 2^n \), where \( n \) is the number of elements in the original set.
The powerset concept is integral in many areas of mathematics and computer science as it encapsulates all possible combinations and arrangements of a set's elements. This provides a comprehensive foundation for exploring functions, relations, and further complex constructs in set theory.

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Most popular questions from this chapter

Use the Venn diagram and the given conditions to determine the number of elements in each region, or explain why the conditions are impossible to meet. \(n(U)=42, n(A)=26, n(B)=22, n(C)=25\) \(n(A \cap B)=17, n(A \cap C)=11, n(B \cap C)=9\) \(n(A \cap B \cap C)=5\)

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

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. \(A \cap \varnothing\)

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

Let $$ \begin{aligned} U &=\\{1,2,3,4,5,6,7\\} \\ A &=\\{1,3,5,7\\} \\ B &=\\{1,2,3\\} \\ C &=\\{2,3,4,5,6\\} \end{aligned} $$ Find each of the following sets. \((A \cup B) \cap(A \cup C)\)
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