## 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:

*f(s) = {s}*for any*s*in**S**

*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.