When mathematicians find a solution to a problem, they are interested in the practicality of being able to actually compute the solution. You have seen in the box following Example that a set having k elements will have 2k subsets. We will pursue this idea.
If a set has 30 elements, then there will be 230 = 1,073,741,824 subsets. If you could write one subset every second, how long would it take you to list all 230 subsets? (Hint: There are 365 x 24 x 60 x 60 seconds in a year—we are ignoring leap years.)
Example Finding All Subsets of a Set Systematically Find all subsets of the set {1, 2, 3, 4}.
We can organize this problem by considering subsets according to their size, going from 0 to 4. This method is illustrated in the following table.
Size of Subset | Subsets of This Size
| Number of Subsets of This Size |
0 | ∅ | 1 |
1 | {1}, {2}, {3}, {4} | 4 |
2 | {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} | 6 |
3 | {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4} | 4 |
4 | {1, 2, 3, 4} | 1 |
5 |
| Total = 16 |
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