2.2. Uncountable Infinity

Examples 2.2.4(b):

The cardinality of the power set of S is always greater than or equal to the cardinality of a set S for any set S.
To show that the cardinality of P(S) is greater than or equal to that of a set S we have to find either a surjection from P(S) to S or an injection from S to P(S).

Define the function f from S to P(S) as follows:

i.e. every element s in S is mapped to the set {s} containing the single element s. Note that the set {s} is an element of the power set P(S).

This map is clearly one-to-one, and therefore card (S) card(P(S)).

This does not prove that card(P(S)) > card(S). However, that statement is also true, but its proof is more complicated.

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