📂Set Theory

Power Set Axiom

Axioms ¹

$$\forall X \exists P \forall A ( A \subset X \implies A \in P)$$ For any set $X$, there exists a set $P$ that contains every subset of $X$ as an element.

Explanation

The power set of $X$ is generally denoted by $\mathcal{P} (X)$ or $2^{X}$, the reason being if the number of elements of a finite set $X$ is denoted by $|X|$, then $\left| \mathcal{P} (X) \right| =2^{|X|}$. It’s not always about the number, so the more someone uses set theory, the more they prefer expressions like $2^{X}$.

One can guess to some extent from the fact that the number of elements increases exponentially, but a power set $2^{X}$ is quite, very large compared to $X$. Moreover, according to the axiom of power set, there also exists a power set of a power set, thus, one can think of sets that grow in size indefinitely in this manner.

As an example of a power set, consider the following:

$$X = \left\{ 1,2 \right\}$$

$$2^{X} = \left\{ \emptyset , \left\{ 1 \right\} , \left\{ 2 \right\}, \left\{ 1,2 \right\} \right\}$$ Note that the empty set $\emptyset$ and the original set $X$ belong to $2^{X}$. The following are some basic properties of power sets. That the empty set is also an element of $2^{X}$ might be awkward at first encounter, but one should naturally accept it since an empty set is also a set and a set can be an element of a set.

Basic Properties

• [1]: $A \subset X \iff A \in 2^{X}$
• [2]: $\emptyset \in 2^{X}$
• [3]: $X \in 2^{X}$

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If one’s concept of set inclusion is weak, it’s naturally confusing. Unfortunately, when one first encounters set theory, power sets are not used extensively and there aren’t particularly good practice problems to get accustomed to. Realistically, one must think time is the remedy and slowly get used to it.