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Chapter 11: Problem 31

Determine the number of subsets of each of the following. A set of 7 elements

Short Answer

Expert verified
128

Step by step solution

01

Understanding Subsets

A subset is any selection of elements from a set, including the empty set and the set itself. If a set has n elements, the number of subsets is given by the formula: Formula: \[ 2^n \]
02

Identifying the Number of Elements

Identify the number of elements in the given set. The set in this exercise has 7 elements, so here, \( n = 7 \).
03

Apply the Formula

Use the formula \[ 2^n \] and substitute \( n = 7 \). \[ 2^7 \]
04

Calculate the Expression

Compute the value: \[ 2^7 = 128 \]. Thus, the number of subsets of a set with 7 elements is 128.

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Key Concepts

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

number of elements
In this section, we'll explore what the 'number of elements' means in a set.
When we talk about a set in mathematics, it is essentially a collection of distinct objects or elements. The 'number of elements' is simply how many individual items are in that set.
For example, in a set A = {1, 2, 3}, there are 3 elements: 1, 2, and 3.
Knowing the number of elements is crucial because it helps us determine other properties, such as the number of subsets the set can have.
subsets formula
Now, let's dive into the formula used to determine the number of subsets.
A subset is a selection of elements from a set, and this can range from the empty set (which includes no elements) to the set itself.
The formula used to calculate the total number of subsets of a set is \((2^n)\), where n represents the number of elements in the set.
Here are some points to remember:
  • The smallest subset is the empty set.
  • The largest subset is the set itself.
  • All possible combinations of elements are considered subsets.
For instance, if we have a set with 7 elements, like in our original exercise, applying the formula \((2^n)\) where \( n = 7 \), gives us \((2^7 = 128)\) subsets.
exponents
To understand the subsets formula better, it's important to grasp the concept of exponents.
An exponent indicates how many times a number, known as the base, is multiplied by itself.
The notation looks like this: \((a^b)\), where \( a \) is the base, and \( b \) is the exponent.
For example, in \((2^3)\), the number 2 is multiplied by itself 3 times: 2 * 2 * 2, which equals 8.
In our exercise, the exponentiation \((2^7)\) means multiplying the number 2 by itself 7 times: 2 * 2 * 2 * 2 * 2 * 2 * 2, resulting in 128.
Hence, using this exponent helps us determine the number of subsets for any given set of elements swiftly.

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