Question

Suppose that \(|A|=m\) and \(|B|=n .\) Find the following cardinalities. $$ |\mathscr{P}(\mathscr{P}(\mathscr{P}(A)))| $$

Short answer

The cardinality of the power set of the power set of the power set for a set \(A\) is \(2^{(2^{2^m})}\.

Step-by-step solution

Calculation of the Cardinality of Power Set \(P(A)\)

The cardinality (or the number of elements) of the power set of set \(A\) denoted as \(P(A)\), will be \(2^m\). The reason for this is because for each element in set \(A\), we have two choices - either we include it in the subset, or we do not. Hence, for \(m\) elements, we have \(2^m\) subsets.

Calculation of the Cardinality of Power set \(P(P(A))\)

The calculation of the cardinality of \(P(P(A))\), is now found in a similar way as in Step 1. However, as the set input to our power set function now is \(P(A)\), the cardinality of this set is \(2^m\). Hence, the cardinality of the set \(P(P(A))\), is now \(2^{2^m}\) which is also \(2^{(2^m)}\) in order to emphasize the step-by-step procedure.

Calculation of the Cardinality of Power set \(P(P(P(A)))\)

Repeating the process once again, the cardinality of the set \(P(P(P(A)))\), where the input set itself has a cardinality of \(2^{(2^m)}\), leading to a cardinality of \(2^{(2^{2^m})}\). Therefore, \(|\mathscr{P}(\mathscr{P}(\mathscr{P}(A)))|=2^{(2^{2^m})}\).

Key concepts

Set Theory

Set theory is a fundamental branch of mathematical logic that studies sets, which are collections of objects. In the realm of set theory, anything can be considered an object, including numbers, characters, or even other sets. At the heart of set theory is the concept of membership; an object either belongs to a set or it does not. This dichotomy is the basic building block for more complex structures in mathematics.

Sets are typically noted by capital letters such as 'A' or 'B', and the notation \( A = \{a, b, c\} \) means that the set A consists of elements a, b, and c. Understanding the nature of sets and their properties lays the groundwork for advanced mathematical topics, including the one tackled in our original problem regarding the calculation of the cardinality of power sets.

Cardinality Calculation

Cardinality refers to the number of elements in a set, which can be finite or infinite. When we're dealing with finite sets, cardinality can simply be counted. For instance, the set \( A = \{1, 2, 3\} \) has a cardinality of 3, often denoted as \( |A| = 3 \).

However, when it comes to power sets, the cardinality calculation isn’t as straightforward. A power set \( \mathscr{P}(A) \) is a set containing all possible subsets of A, including the empty set and A itself. The fascinating part of power sets is their size: for a set with cardinality \( m \), its power set has a cardinality of \( 2^m \) - a result derived from the idea that every element can either be included or excluded in a subset, leading to two possibilities for each element. This exponential growth is core to understanding the operation of power sets, as seen in the textbook solution.

Elements of a Set

When we talk about the 'elements of a set', we're specifically referring to the distinct objects that make up the set. Each element in a set is unique; duplicates are not allowed within the standard mathematical definition of a set. The elements can be anything from numbers to symbols to other sets, which brings an interesting point in our example.

For power sets, the elements themselves are sets, as every subset of the original set is an element of the power set. When looking at higher-order power sets like \( \mathscr{P}(\mathscr{P}(\mathscr{P}(A))) \), the elements become sets of sets of sets. Here, appreciating the recursive nature of elements within power sets is crucial to grasping the more complex concepts of set theory, such as those highlighted in the steps to calculate cardinalities in our exercise.