Homework 4: Due Friday, 29 October at 5PM

1.) Let f and g be functions of one argument.  The composition of
    f and g is the function x :--> f(g(x)).  Define a procedure 
    compose that takes two functions and implements composition.  For
    example, ((compose square square) 2) --> 16.

2.) Exercise HTDP 21.1.2 (p. 310)

3.) Since we have functions, we really don't need structures or lists
    as primitives in our language. To see this:

(define (make-pair a b) (lambda (m) (m x y)))
(define (pair-a pr) (pr (lambda (p q) p)))

  a.) Explain these definition, and verify that (pair-a (make-pair a b))
      indeed returns a by substitution.
  b.) Give an implementation of pair-b that extracts the second item
      from a pair.

4.) Since we have functions, we don't need numbers (at least nonnegative
    integers) as primitives in our language.  Take the definitions

(define zero (lambda (f) (lambda (x) x)))
(define (add-1 n)
  (lambda (f) (lambda (x) (f ((n f) x)))))

  a.) Explain the meaning of zero and add-1.
  b.) Define one and two directly (without using zero or add-1).
  c.) Give a direct (nonrecursive) definition of addition.
  d.) Give a direct (nonrecursive) definition of multiplication.
  e.) (Bonus points) Give a definition of (sub-1 n) that subtracts
      one from any non-zero number.