Why the set of all subsets of a finite set $X$ is denoted by $2^X$? Why the set of all subsets of a finite set $X$ is denoted by $2^X$ ? specifically why the number 2? what does it represents?
Could anyone explain this for me please? 
 A: Consider each element of the set $x \in X$. For every subset $X' \subset X$ we can ask the question is $x$ in $X'$? The answer will always be no or yes so there are two possible outcomes for each $x$ and lets name them $\{0,1\}$. This means for each subset we can define a function $f_{X'}:X \rightarrow \{0,1\}$. Since we have all possible subsets we also have all possible functions $f:X \rightarrow \{0,1\}$ if we consider every possible $f_{X'}$, so there is a bijection between the subsets of $X$ and the set of all functions from $X$ to $\{0,1\}$.
So the reason we use $2^X$ is because the set of all functions from $X$ to $Y$ is $Y^X$ and since $\{0,1\}$ has two elements we simply use the number $2$ as a shorthand. It also draws attention to the fact that $|Y^X|=2^{|X|}$ for finite set so it's a useful notation as well.
A: Because if the set $X$ has $n$ elements, then the power set of $X$ has $2^n$ elements.
$$ |X| = n \implies  |\mathcal{P}(X)| = 2^n$$
A: In general, $A^B$ means the set of all functions from $B$ to $A$, and its cardinality is $|A^B| = |A|^{|B|}$. With that in mind, note that each subset $A\subseteq X$ is characterized by its, well, characteristic function $\chi_A\colon X \to 2 = \{0,1\}$ (note/recall that $2 \doteq \{0,1\}$ is a set-theoretic definition of "$2$", when constructing $\mathbb{N}$), given by $$\chi_A(x) = \begin{cases} 0, & \mbox{ if }x\not\in A \\ 1, & \mbox{ if }x \in A \end{cases}$$In other words, the map $$\wp(X) \ni A \mapsto \chi_A \in 2^X$$is a bijection. This is why people write $2^X$ instead of $\wp(X)$ sometimes: because when dealing with sets, these two things are equivalent.
A: Take that a set has $n$ elements. Each element can be or not be an element of any subset. $2$ is coming from these two options: to be or not to be. The rest is just a combination of all options. As each element can be or not be an element of a subset, and each subset is defined by its content, we have $2^n$ subsets.
