Problem 31 - Discrete Mathematics with Applications

Question

Find the power set of each set. $$ \mathrm{~ \\{ a \\} ~} $$

Step-by-Step Solution

Answer

The power set of the set {a} is \(\{ \emptyset, \{a\} \}\).

Step 1: Understanding Subsets

A subset is a set whose elements all belong to another set, called the "parent" set. A set can have many subsets, including the empty set and the parent set itself.

Step 2: Listing Subsets of the Given Set

In this case, the parent set is {a}. We can list the subsets of {a} as follows: 1. The empty set, denoted by ∅ or {}. 2. The parent set itself, which is {a} in this case.

Step 3: Organizing Subsets into the Power Set

Once we have found all the subsets of the given set ({a}), we can now organize them into a new set called the power set. In this case, the power set will contain two elements: the empty set and the parent set itself. The power set of {a} is: $$ \mathcal{P}(\{a\}) = \{ \emptyset, \{a\} \} $$ So, the power set of the set {a} is \(\{ \emptyset, \{a\} \}\).

Key Concepts

Subsets

When we talk about subsets, we are referring to sets that contain elements that all belong to another set.
For example, if we have a set \( S \), any subset should include elements only from \( S \). Subsets can range from the empty set to the set itself.
To put it simply, a subset is like a smaller or equal version of a parent set.

When forming subsets, consider:

• Each combination of the parent set's elements forms a subset.
• Every set includes at least the empty set and itself as subsets.

Understanding subsets is crucial because they are building blocks for the power set, which contains all possible subsets of a particular set.

Empty Set

The concept of the empty set might seem abstract, but it plays a critical role in set theory.
Imagine a set with no elements inside. That's an empty set, often denoted by \( \emptyset \) or \( \{ \} \).
It's like an empty container that still holds significance mathematically.

Key points about the empty set include:

• The empty set is a subset of every set, no matter what.
• Despite having no elements, it is included in the power set of any set.

The presence of the empty set in a power set assures that all possible combinations, including none, are considered.

Parent Set

The parent set acts as the foundation for forming subsets.
This is the original set from which all subsets are derived.
Understanding the parent set helps in visualizing how subsets form and how the power set is constructed.

Consider these thoughts on parent sets:

• The parent set itself is always a subset.
• The number of subsets, including the empty set and the parent set, can be calculated as \( 2^n \), where \( n \) is the number of elements in the parent set.

The parent set is essential in understanding the vastness and structure of the power set.