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Chapter 11: Problem 32

Determine the number of subsets of each of the following. A set of 6 members

Short Answer

Expert verified
The set with 6 members has 64 subsets.

Step by step solution

01

Understand the Set

First, recognize that the set has 6 members. Let鈥檚 denote this set as set A where \(A = \{a_1, a_2, a_3, a_4, a_5, a_6\} \).
02

Formula for Number of Subsets

Use the formula for finding the number of subsets of a set. A set with n elements has \(2^n \) subsets. Here, n = 6.
03

Apply the Formula

Substitute n with 6 in the formula: \(2^n \) subsets. That yields \(2^6 \).
04

Calculate the Number of Subsets

Compute \(2^6 = 64 \). So the number of subsets is 64.

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Key Concepts

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

Understanding Subsets
A subset is a set that contains some or all elements of another set. For example, if we have a set A with members \( A = \{a_1, a_2, a_3, a_4, a_5, a_6\}\) , then a subset of A might be \( B = \{a_1, a_4\}\), where B includes elements from A. The concept of subsets is fundamental in set theory. Here's why: subsets allow us to explore and understand the number of ways we can choose elements from a set. Each element can either be in the subset or not. This yes/no choice is the cornerstone of subsets.
Determining the Number of Elements
The number of elements in a set is simply how many items are in the set. In this example, we have a set A with 6 elements: \( A = \{a_1, a_2, a_3, a_4, a_5, a_6\} \). Counting elements is essential because it helps us understand the size of the set. Sets can be of any size: finite or infinite. When we mention a set with 6 members, n = 6, where n represents the total number of elements. Having this number is crucial for further calculations and understanding the scope of the set.
Formula for Subsets
To find the number of subsets of a set, we use a formula from combinatorics: \( 2^n \), where n is the number of elements in the set. This formula works because each element in the set has two choices: either to be included in a subset or not. Let's break it down:
  • If a set has 1 element, it has \( 2^1 = 2 \) subsets (the set itself and the empty set).
  • If a set has 2 elements, it has \( 2^2 = 4 \) subsets.
  • If a set has 3 elements, it has \( 2^3 = 8 \) subsets.
Following this pattern, our set with 6 elements has \( 2^6 = 64 \) subsets. This showcases the exponential growth of subsets as the number of elements increases.
Introduction to Combinatorics
Combinatorics is the branch of mathematics that deals with counting, arranging, and combining objects. It's a key area in set theory, helping us determine the number of ways we can choose or arrange elements. For example, calculating the number of subsets is a combinatorial problem because we count combinations of elements. The basic principles of combinatorics include the Rule of Product (multiplying possibilities), the Rule of Sum (adding possibilities), and combinations (selecting items without considering order). When dealing with sets, combinatorics helps us find efficient ways to count possibilities, such as the number of subsets.

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Most popular questions from this chapter

Prove that $$ \left(\begin{array}{c} n \\ k-1 \end{array}\right)+\left(\begin{array}{l} n \\ k \end{array}\right)=\left(\begin{array}{c} n+1 \\ k \end{array}\right) $$ for any natural numbers \(n\) and \(k, k \leq n\)

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