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Chapter 2: Problem 104

Describe how to find the number of distinct proper subsets for a given set. Give an example.

Short Answer

Expert verified
The number of distinct proper subsets of a set is calculated with \((2^n) - 1\). E.g, for the set A = {1, 2, 3}, the number of distinct proper subsets will be (2^3) - 1 = 7.

Step by step solution

01

Understanding Proper Subsets

In Set Theory, a Proper subset of a set A is a subset that is not the same as set A. In other words, if set B is a proper subset of set A, then all elements of B are in A but A contains at least one element not contained in B.
02

Finding the Total Number of Subsets

The total number of subsets (including both proper and improper) of a set with n elements is given by \(2^n\). This is because each element can either be in the subset or not, giving us 2 choices for each element, hence \(2^n\) total choices.
03

Calculating the Number of Proper Subsets

The number of proper subsets is gotten by subtracting 1 from the total number of subsets. This is because we subtract the one case where the subset is the same as the original set (improper subset). The formula becomes \(2^n - 1\).
04

Applying the Formula to an Example

If we have a set A = {1, 2, 3}, the number of distinct proper subsets would be calculated as follows: first, find the number of elements in A (n). In this case, n=3. So the total number of subsets is \(2^n = 2^3 = 8\). Then subtract 1 to determine the number of proper subsets: 8 - 1 = 7. Therefore, the number of distinct proper subsets of A is 7.

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Key Concepts

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

Proper Subset
In set theory, the concept of a proper subset is fundamental. A proper subset of a set \(A\) is a subset that contains some, but not all, elements of \(A\). Importantly, it cannot be equal to \(A\). For instance, if \(A = \{1, 2, 3\}\), then \(\{1, 2\}\) is a proper subset of \(A\). However, \(\{1, 2, 3\}\) is not, because it is equal to \(A\). The distinction is that a proper subset never includes all elements of the original set, ensuring it remains 鈥榩roperly鈥� smaller.
Number of Subsets
Determining the number of subsets, both proper and improper, is straightforward with a set theory formula. If a set \(A\) contains \(n\) elements, the total number of possible subsets is given by \(2^n\). This arises because each element can either be included or excluded from a subset, leading to two choices. Thus, for a set with 3 elements like \(\{1, 2, 3\}\), there are \(2^3 = 8\) possible subsets, ranging from an empty set to the set itself.
Mathematical Example
Let's take a set \(A = \{1, 2, 3\}\) as an example. To find the number of distinct proper subsets, we first compute the total subsets using the formula \(2^n\). Here, \(n = 3\), so \(2^3 = 8\). Since one of these is the set itself, \(\{1, 2, 3\}\), which is improper, we subtract it from the total, yielding \(8 - 1 = 7\) proper subsets. Thus, the proper subsets include variations like:
  • \(\{1\}\)
  • \(\{2\}\)
  • \(\{3\}\)
  • \(\{1, 2\}\)
  • \(\{1, 3\}\)
  • \(\{2, 3\}\)
  • \(\{\}\) (the empty set)
Subset Formula
The formula for calculating the number of proper subsets of a set takes into account the total subsets and excludes one. The total subsets are given by \(2^n\), where \(n\) is the number of elements. To find the number of proper subsets, simply subtract 1, because we do not count the subset that is equal to the original set. Therefore, the formula becomes:\[2^n - 1\]This provides a quick way to find the proper subsets, as demonstrated with our example, where \(2^3 - 1 = 7\). Understanding this formula helps in efficiently determining the proper subsets for any given set.

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Most popular questions from this chapter

This group activity is intended to provide practice in the use of Venn diagrams to sort responses to a survey. The group will determine the topic of the survey. Although you will not actually conduct the survey, it might be helpful to imagine carrying out the survey using the students on your campus. a. In your group, decide on a topic for the survey. b. Devise three questions that the pollster will ask to the people who are interviewed. c. Construct a Venn diagram that will assist the pollster in sorting the answers to the three questions. The Venn diagram should contain three intersecting circles within a universal set and eight regions. d. Describe what each of the regions in the Venn diagram represents in terms of the questions in your poll.

Describe the Venn diagram for two disjoint sets. How does this diagram illustrate that the sets have no common elements?

Describe the Venn diagram for two sets with common elements. How does the diagram illustrate this relationship?

A small town has four police cars. If a radio dispatcher receives a call, depending on the nature of the situation, no cars, one car, two cars, three cars, or all four cars can be sent. How many options does the dispatcher have for sending the police cars to the scene of the caller?

If \(A \cap B=\varnothing\), then \(n(A \cup B)=n(A)+n(B)\).
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