Problem 29 - Algebra for College Students

Question

Make a list of all of the subsets of the set \(\\{a, b, c\\} .\) How many are there?

Step-by-Step Solution

Answer

The subsets are \(\{ \}\), \(\{a\}\), \(\{b\}\), \(\{c\}\), \(\{a, b\}\), \(\{a, c\}\), \(\{b, c\}\), and \(\{a, b, c\}\). There are 8 subsets.

Step 1 - Understand Subsets

A subset is a set that contains some or all elements of another set. The empty set and the set itself are always subsets.

Step 2 - Identify the Original Set

We are given the set \(\{a, b, c\}\). We will list all possible subsets.

Step 3 - List the Empty Set

The first subset to list is the empty set: \(\{ \}\)

Step 4 - List Subsets with One Element

Next, list all subsets containing one element: \(\{a\}\), \(\{b\}\), \(\{c\}\)

Step 5 - List Subsets with Two Elements

Now, list all subsets containing two elements: \(\{a, b\}\), \(\{a, c\}\), \(\{b, c\}\)

Step 6 - List the Subset with All Elements

Finally, list the subset containing all elements: \(\{a, b, c\}\)

Step 7 - Count the Subsets

Count the number of subsets listed: \(\{ \}\), \(\{a\}\), \(\{b\}\), \(\{c\}\), \(\{a, b\}\), \(\{a, c\}\), \(\{b, c\}\), \(\{a, b, c\}\). There are 8 subsets in total.

Key Concepts

set theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, known as elements. These objects could be anything like numbers, letters, or even other sets. A set is usually denoted by curly braces \( \{ \} \) with elements separated by commas. For example, the set \( \{a, b, c\} \) contains three elements: a, b, and c.

Sets can be categorized into different types, such as finite, infinite, countable, and uncountable sets. Set theory helps in understanding and organizing collections of objects in a structured way. This is especially useful when dealing with problems in mathematics, computer science, and many other fields.

Basic operations in set theory include Union (\(A \cup B\)), Intersection (\(A \cap B\)), and Difference (\(A - B\)). These operations allow for combining sets in various ways to form new sets. The concept of subsets is a vital part of set theory.

subsets

A subset is a set that consists of elements from another set. If set A is a subset of set B, all elements of A are also elements of B. This can be written as \(A \subseteq B\).

Every set has at least two subsets: the empty set and the set itself. For example, for the set \( \{a, b, c\} \), both \( \{ \} \) (empty set) and \( \{a, b, c\} \) (the set itself) are subsets.

To list all subsets of a set with 'n' elements, consider every possible combination of those 'n' elements. Each element can either be included or excluded from a subset. So, if we have a set \(S\) with 'n' elements, the total number of possible subsets is \(2^n\). For instance, the set \( \{a, b, c\} \) has 3 elements, and therefore has \(2^3 = 8\) subsets.

counting subsets

Counting subsets involves determining the total number of subsets a particular set has. As mentioned, if a set has 'n' elements, the number of subsets is given by \(2^n\). This formula comes from the fact that each element has two possibilities: either it is included in a subset, or it is not.

Let's use the set \( \{a, b, c\} \) as an example. This set has 3 elements. Using the formula \(2^n\), we find: \[ 2^3 = 8 \]

This means there are 8 subsets. Counting them, we get:

• Empty set: \( \{\} \)
• One-element subsets: \( \{a\}, \{b\}, \{c\} \)
• Two-element subsets: \( \{a, b\}, \{a, c\}, \{b, c\} \)
• Three-element subset: \( \{a, b, c\} \)

These combinations cover all possible subsets, confirming our calculation.

empty set

The empty set, often denoted as \( \{\} \) or \( \emptyset \), is a unique set that contains no elements. It is a fundamental concept in set theory and is considered a subset of every set.

For any set, the empty set is always one of its subsets. This is because there isn't an element in the empty set that isn't also in the other set, satisfying the subset condition: all (zero) elements of the empty set are contained in every other set.

The empty set is crucial for various mathematical operations and proofs. It serves as a foundation for defining other sets and understanding the relationships between them.

In our example, \( \{a, b, c\} \) includes the empty set as one of its subsets, which is always important to count when listing all possible subsets.