How do you find all the subsets of a set using a recursive approach?

Finding All Subsets of a Set

Given a set of n elements, a subset is any combination of those elements. The goal is to find a comprehensive algorithm that generates all possible subsets.

Recursive Solution

Consider the following algorithm:

• Base Case: If the set contains only one element, the algorithm returns an empty set and a set containing that element.
• Recursive Step: For a set with n elements, find the set of subsets of the first n-1 elements.
• Split: Divide the previous set into two groups: subsets that contain the nth element and subsets that do not.
• Union: Take the union of these two groups to form the complete set of subsets.

Example: {1,2,3,4,5}

Step 1: Find all subsets of {1,2,3,4}. These are: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, and {1,2,3,4}.

Step 2: Add 5 to each subset from Step 1 and union with the subsets:

• {} -> {}
• {1} -> {1} and {1,5}
• {2} -> {2} and {2,5}
• ...
• {1,2,3,4} -> {1,2,3,4} and {1,2,3,4,5}

The union of these subsets gives us the complete set of subsets for {1,2,3,4,5}:

{ {}, {1}, {2}, {3}, {4}, {5}, {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}, {1,2,3}, {1,2,4}, {1,2,5}, {1,3,4}, {1,3,5}, {1,4,5}, {2,3,4}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,4}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, and {1,2,3,4,5} }

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