What does $2^X$ mean in $2^X:= {\mathcal {P}}(X)$

Question

wiki gives this definition of sigma-algebra

Let X be some set, and let ${\mathcal {P}}(X)$ represent its power set. Then a subset ${\displaystyle \Sigma \subseteq {\mathcal {P}}(X)}$ is called a σ-algebra if it satisfies the following three properties:

1. X is in Σ, and X is considered to be the universal set in the following context.
2. Σ is closed under complementation: If A is in Σ, then so is its complement, X \ A.
3. Σ is closed under countable unions: If $A_1, A_2, A_3, ...$ are in Σ, then so is $A = A_1 ∪ A_2 ∪ A_3 ∪ …$ .

since {X, ∅} satisfies condition (3), it follows that {X, ∅} is the smallest possible σ-algebra on X. The largest possible σ-algebra on X is $2^X:= {\mathcal {P}}(X)$

$2^X$ does not seem to be 2 to the Xth power here, so, What does $2^X$ mean in $2^X:= {\mathcal {P}}(X)$?

Answer 1

Let's discuss a more general notion: the notation $A^B$ for sets $A,B$. This is defined as the set of functions from $B$ to $A$, i.e.

$$A^B := \{ f \mid f : B \to A \}$$

If you're familiar with, say, the set-theoretic construction of $\Bbb N$, you will recall that each $n \in \Bbb N$ can be identified loosely as some collection of nested sets of the empty set, in the sense that $0 = \emptyset$ and the successor function gives $\sigma(n+1) = \{n\} \cup n$. Thus we can identify $1$ with $\{ \emptyset \}$, $2$ with $\{ \emptyset , \{ \emptyset \}\}$, and so on. Each $n \in \Bbb N$ is thus identified with a set of cardinality $n$ in this manner.

Note: You can also replace the $\{ \emptyset \}$ in $2$'s set with $1$ since $1 = \{ \emptyset \}$. This gives you the equivalent $2 = \{0,1\}$ Arturo noted in the comments. You can thus get the more compact and perhaps more intuitive notation of $n = \{0,1,2,\cdots,n-1\}$ for any $n \in \Bbb N$ like this.

So then, let $A=2, B=X$. What is $2^X$? By the previous, $2^X$ is the set of functions from $X$ to the set of two elements. How can we identify this with the power set $\mathcal P(X)$, the set of subsets of $X$?

Simple. Identify a $f \in 2^X$ as the indicator function: let $f(x) = 1$ if $x \in Y$ ($Y$ being a particular, chosen subset of $X$), and $0$ otherwise. This means, in other words, we can identify $f$ uniquely with the subset of $X$ containing the elements $f$ sends to $1$. The elements sent to $0$ likewise denote the elements not in that particular subset.

Some quick examples:

• If $f(x) = 0$ for all $x\in X$, then $f$ is identified with the empty set, which of course is a subset of $X$.
• Similarly, if $f(x) = 1$ for all $x \in X$, $f$ is identified with $X$ itself, since $X \subseteq X$.
• If $f(x) = 1$ for only one $x \in X$, then $f$ is identified with $\{x\}$, i.e. the set containing only $x$.

And so on and so forth.

With each $f \in 2^X$ uniquely identified with some $Y \in \mathcal P(X)$, then we can conclude $2^X = \mathcal P(X)$.

Answer 2

Given a set S = {x, y, z}, then all the possible subsets of S are

{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}

where, {} denotes the empty set (also denoted $\varnothing$).

The power set of S is a set containing all possible subsets of S, including the empty set and S itself, that is, {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}.

$2^S$ denotes the power set of S.

If S is a finite set with |S| = n elements (3, in this example), then the number of all the possible subsets of S is $2^n$ (8, in this example), which is the motivation to use $2^S$ notation.

as for your question, $2^$ is the notation of the power set of X

more info is here.