Cardinality and the Properties of a Power Set

What is the purpose of a power set? You’ve probably employed the intuition underpinning power sets without even realising it. When you select a subset of items from a bigger set, you are picking a power set item. For example, if a child has $5 and wants to go shopping for candy, which aspect of the power set of all accessible candies will she choose? For the more technically inclined, as software engineers, you would wish to query all potential database users who also have properties X & Y – another example of selecting one subset from all possible subsets.

Definition of Power Set

All subsets of a given set, including the empty set, are included in a power set. The power set is represented by the notation P(S), and the number of elements is provided by 2n. A power set may be thought of as a container for all the subsets of a given set; in other words, the members or elements of a power set are the subsets of a set.

In simple terms, a set is a collection of different items. If there are two sets A and B, then set A is the subset of set B if all of set A’s items are present in set B. With the assistance of examples, let us learn more about the qualities of a power set, the cardinality of a power set, and the power set of an empty set.

Properties of Power Set

Power sets are sets that contain a list of all the subsets of a particular set; they are also known as power sets. Among the characteristics of the power set indicated by P(A) with ‘n’ members are the following ones:

A set has n elements in total, which is the number of items in the set.

An empty set is a distinct constituent of a power set, but a full set is not.

There is just one element in the power set of an empty set.

The power set of a set with a finite number of items also has a finite number of elements. Power sets are countable, for example,  X = {b,c,d}

There are an unlimited number of subsets in the power set of an infinite set. Consider the following scenario: If Set X contains all of the multiples of 5 beginning with 5, then we may claim that Set X has an infinite number of items. Despite the fact that set X has an unlimited number of items, a power set exists for it, and in this instance, it contains an endless number of subsets.

These the power set exists for both finite and infinite sets.

The Cardinality of Power Set

The number of different items inside a set, often known as its cardinality, serves as a basic jumping-off point for an additional, fuller study of a particular set. For one thing, cardinality is the first unique attribute we’ve encountered that allows us to objectively compare various sorts of sets — evaluating if a bijection (fancy name for function with minor qualifiers) exists from one set to another. Another type of application, and the subject of the rest of this article, cardinality gives a window into all the subsets that exist within a particular set. This extends directly to common choice allocation concerns like planning a supermarket trip or managing a portfolio.

A set’s cardinality is the total number of items in the set. A power set is a collection of all the subsets of a set. 2n gives the total number of subsets for a set of ‘n’ items. Because the elements of a power set are subsets of a set, the cardinality of a power set is given by |P(A)| = 2n. In this case, n represents the total number of elements in the provided set.

Example: Set A = {1,2}; n = 2

|P(A)| = 2n = 2.2 = 4.

Subsets of A = {}, {1},{2},{1,2}

Therefore, |P(A)| = 4.

Conclusion

With basic notation and operations covered in parts one and two of this series, we now have a solid foundation of Set Theory. This third article expands on this information by focusing on the most crucial attribute of every given set: the total number of distinct members it includes.