# Interactive Real Analysis

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## 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
Hence:
• P(S) = {0, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, S}
Therefore, we have:
• card(S) = 3 and card(P(S)) = 8 = 23
For the set 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
Therefore, we have:

• card(S) = 4 and card(P(S)) = 16 = 24
Finally, if S = {1,2,3,4,5,6} then, based on the above examples, we would suspect that

• card(S) = 6, therefore card(P(S)) = 26 = 64
In fact, if a set S contains n elements, then its power set will contain 2n elements. This can be proved by induction as an exercise.
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