Explanation: Connection Between the Power Set and Functions Mapping Sets into a Set with Two Elements

(related to Part: Set-theoretic Prerequisites Needed For Combinatorics)

The indicator function is an example of a map of an arbitrary set $X$ into another set with two elements, in this case, the set $\{0,1\}$. The definition of indicator function reveals that all maps of this kind are exactly those maps, whose carriers correspond to exactly the subsets of $X.$

The question, how many characteristic functions for a given set $X$ do exist, is therefore equivalent to the question, how many subsets of $S$ exist. But this is exactly the cardinality $|\mathcal P(X)|,$ where $\mathcal P(S)$ denotes the power set of $S.$ This result can be summarized as follows: $$|\mathcal P(X)|=2^{|X|}.$$ It is true for both, finite or infinite sets $X.$ In the combinatorics, $2^{|X|}$ is the common notation for the cardinality of the power set $\mathcal P(S).$

Despite this argument, the following proposition summarizes this result for finite sets with yet another proof:

  1. Proposition: Number of Subsets of a Finite Set

Proofs: 1

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  1. Flachsmeyer, Jürgen: "Kombinatorik", VEB Deutscher Verlag der Wissenschaften, 1972, 3rd Edition