Problem 2 - A Gentle Introduction to the Art of Mathematics

Question

Try iterating the power set operator. What is \(\mathcal{P}(\mathcal{P}(\emptyset)) ?\) What is \(\mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) ?\)

Step-by-Step Solution

Answer

The power sets are: \( \mathcal{P}(\mathcal{P}(\emptyset)) = \{ \emptyset, \{ \emptyset \} \} \) and \( \mathcal{P}(\mathcal{P}(\mathcal{P}(\emptyset))) = \{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \} \).

Step 1: Understanding the Problem

We need to find the iterated power set of the empty set. This means repeatedly applying the power set operator to the empty set \( \emptyset \).

Step 2: First Iteration: Power Set of the Empty Set

The power set of the empty set, \( \mathcal{P}(\emptyset) \), is the set of all subsets of the empty set. The only subset of the empty set is the empty set itself. Therefore, \( \mathcal{P}(\emptyset) = \{ \emptyset \} \).

Step 3: Second Iteration: Power Set of \{ \emptyset \}

Next, we find the power set of \( \mathcal{P}(\emptyset) = \{ \emptyset \} \), which is \{ \emptyset \} \, as follows. The subsets of \( \{ \emptyset \} \) are \emptyset \ and \{ \emptyset \} \. Thus, \( \mathcal{P}( \{ \emptyset \} ) = \{ \emptyset, \{ \emptyset \} \} \).

Step 4: Third Iteration: Power Set of \{ \emptyset, \{ \emptyset \} \}

Finally, we find the power set of \( \{ \emptyset, \{ \emptyset \} \} \). The subsets of \{ \emptyset, \{ \emptyset \} \} \ are: \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, and \{ \emptyset, \{ \emptyset \} \} \. Thus, \( \mathcal{P}( \{ \emptyset, \{ \emptyset \} \} ) = \{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \} \).

Key Concepts

Empty Set

The empty set, also known as the null set, is a unique set in set theory. It is denoted by \(\boldsymbol{\emptyset}\).
It is unique because it contains no elements. For example, if we have a set for people who are currently on the moon, it would be empty since there are no people there right now.
In mathematical notation, we say that \(\boldsymbol{\emptyset}\) has zero elements: \(\boldsymbol{|\emptyset| = 0}\).

Set Theory

Set theory is the branch of mathematical logic that deals with sets, which are collections of objects. Sets can contain numbers, letters, or other objects, and they are generally denoted by curly braces.
For example, \(\{a, b, c\}\) is a set containing elements \'a\', \'b\', and \'c\'.
Set operations, such as union, intersection, and power sets, are fundamental to understanding set theory.
Set theory is foundational to many areas of mathematics.
The power set, denoted by \(\boldsymbol{\mathcal{P}(S)}\), is particularly important. It is the set of all subsets of a set \(\boldsymbol{S}\).
For instance, if \(S = \{1, 2\}\), then \(\boldsymbol{\mathcal{P}(S)} = \{ \emptyset, \{1\}, \{2\}, \{1, 2\} \}\).

Subsets

A subset is a set whose elements are all contained within another set.
For example, if we have a set \(A = \{1, 2, 3\}\), the sets \( \{1\}, \{2\}, \{1, 2\}\) are all subsets of \(\boldsymbol{A}\).
We say that \(\boldsymbol{B}\) is a subset of \(\boldsymbol{A}\) if every element of \(\boldsymbol{B}\) is also an element of \(\boldsymbol{A}\), denoted by: \(\boldsymbol{B \subseteq A}\).
A set is always a subset of itself, and the empty set is a subset of every set.
Understanding subsets is crucial for the concept of the power set.