SICP Exercise 2.32 subsets

Exercise 2.32.  We can represent a set as a list of distinct elements, and we can represent the set of all subsets of the set as a list of lists. For example, if the set is (1 2 3), then the set of all subsets is (() (3) (2) (2 3) (1) (1 3) (1 2) (1 2 3)). Complete the following definition of a procedure that generates the set of subsets of a set and give a clear explanation of why it works:

(define (subsets s)
  (if (null? s)
      (list nil)
      (let ((rest (subsets (cdr s))))
        (append rest (map <??> rest)))))

 SOLUTION
The code and tests are here.

Explanation: This problem is identical to the count-change problem that appears earlier in this book. Let's say that from a given set S, we pick any element e. Then, the elements in the set S' of all subsets of S can be segregated into two groups: one whose members contain e and the other whose members do not contain e.

 The logic used here is as follows: The set S' of all subsets of a given set S is the union of the following two sets:

 1. The set of all subsets of the set obtained after removing one element, say e from S. Let's call this set P.
2. The set Q generated from P by adding e to each element in P

 The 'map' function above does exactly what is described in  step 2 above. The variable 'rest' is the same as set P above.

 S' = P union Q

 Note: The procedure above works well even when the set elements are lists/pairs themselves

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