Problem 2 List all subsets of the followin... [FREE SOLUTION]

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Applied Finite Mathematics

Rupinder Sekhon

$Math Studyset 91影视 Explanations$ Math

2011 Edition

Chapter 12: Problem 2

List all subsets of the following set. $$ \\{\mathrm{Al}, \text { Bob, Chris }\\} $$

Short Answer

Expert verified

The subsets of the set \{Al, Bob, Chris\} are: $\emptyset$, \{Al\}, \{Bob\}, \{Chris\}, \{Al, Bob\}, \{Al, Chris\}, \{Bob, Chris\}, \{Al, Bob, Chris\}

Step by step solution

List the Null Subset

Start by listing the null or empty set. This is a subset which contains no elements. It's written as $\emptyset$ or \{\}

List the Single-Element Subsets

Next, list all the subsets that consist of a single element from the original set. There will be as many of these as there are elements in the given set. These subsets are: \{Al\}, \{Bob\}, \{Chris\}

List the Two-Element Subsets

After that, list out all the subsets that consist of two elements from the original set. Combine each element with every other element without repeating the combinations. The subsets are: \{Al, Bob\}, \{Al, Chris\}, \{Bob, Chris\}

List the Three-Element Subset

Finally, there's the subset that contains all elements from the original set, which is \{Al, Bob, Chris\} itself.

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Key Concepts

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

Null Subset

In mathematics, every set, no matter how many elements it has, includes a special subset called the null subset. Also known as the empty set, this subset contains no elements at all. Think of it as the blank page in a book, something that exists but is devoid of content. We represent it using the symbols $\emptyset$ or $\{\}$.
Understanding the null subset is crucial because it's a foundational component in set theory. It may seem trivial, but recognizing its existence is key to understanding subsets in mathematics.

Single-element Subsets

Single-element subsets are formed by taking each element from the set and turning it into its own subset. For a set like $\{\text{Al, Bob, Chris}\}$, each of these names can stand alone in a subset. Thus, the single-element subsets are:

$\{\text{Al}\}$
$\{\text{Bob}\}$
$\{\text{Chris}\}$

Viewing single-element subsets helps you see how each item in the original set can independently form its own subset.
This means the number of single-element subsets is equal to the number of elements in the original set. This step is simple, but it helps to build up to more complex subsets.

Two-element Subsets

Two-element subsets include all possible combinations of two different elements from the set. For instance, if you have a set with three elements, you pair them together in every possible way. With $\{\text{Al, Bob, Chris}\}$, the two-element subsets are:

$\{\text{Al, Bob}\}$
$\{\text{Al, Chris}\}$
$\{\text{Bob, Chris}\}$

This process of systematic pairing without repetition ensures that every potential duo is represented. By understanding this concept, you see how each element can work with every other element, creating smaller groups within the set.

Three-element Subset

When a subset contains all the elements of the original set, it is called a three-element subset for a set containing three members. In our example of $\{\text{Al, Bob, Chris}\}$, the three-element subset isn鈥檛 a surprise鈥攊t's the original set itself:

$\{\text{Al, Bob, Chris}\}$

Although it might seem like you're repeating the original set, recognizing it as a subset is important. This is because any set is considered a subset of itself.
This affirms the idea that subsets range from being completely empty (the null set) to entirely full (the original set). Emphasizing every subset variation helps fulfill the exploration of subsets in mathematics.

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91影视

List all subsets of the following set. $$ \\{\mathrm{Al}, \text { Bob, Chris }\\} $$

Short Answer

Step by step solution

List the Null Subset

List the Single-Element Subsets

List the Two-Element Subsets

List the Three-Element Subset

Key Concepts

Null Subset

Single-element Subsets

Two-element Subsets

Three-element Subset

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