## 2.2. Uncountable Infinity

### Examples 2.2.4(a):

If S = {1,2,3}, then what is P(S) ? What is the power set of the set
S = {1, 2, 3, 4} ? How many elements does the power set of
S = {1, 2, 3, 4, 5, 6} have ?

The power set is the set of all subsets of a given set.
For the set **S** = {1,2,3} this means:

- subsets with 0 elements:
**0**(the empty set) - subsets with 1 element: {1}, {2}, {3}
- subsets with 2 elements: {1,2}, {1,3}, {2,3}
- subsets with 3 elements:
**S**

**P**(**S**) = {**0**, {1}, {2}, {3}, {1,2}, {1,3}, {2,3},**S}**

*card(S) = 3*and*card(P(S)) = 8 = 2*^{3}

**S**= {1,2,3,4} this means:

- subsets with 0 elements:
**0**(the empty set) - subsets with 1 element: {1}, {2}, {3}, {4}
- subsets with 2 elements: {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}
- subsets with 3 elements: {1,2,3}, {1,2,4}, {1,3,4} {2,3,4}
- subsets with 4 elements:
**S**

*card(S) = 4*and*card(P(S)) = 16 = 2*^{4}

*card(S) = 6*, therefore*card(P(S)) = 2*= 64^{6}

**S**contains

*n*elements, then its power set will contain

*2*elements. This can be proved by induction as an exercise.

^{n}