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Exercises

## Exercise 2.4

Here is an alternative procedural representation of pairs. For this representation, verify that `(car (cons x y))` yields `x` for any objects `x` and `y`.

``````(define (cons x y)
(lambda (m) (m x y)))

(define (car z)
(z (lambda (p q) p)))``````

What is the corresponding definition of `cdr`? (Hint: To verify that this works, make use of the substitution model of section 1.1.5.)

## Exercise 2.5

Show that we can represent pairs of nonnegative integers using only numbers and arithmetic operations if we represent the pair a and b as the integer that is the product 2a 3b. Give the corresponding definitions of the procedures `cons`, `car`, and `cdr`.

## Exercise 2.6

In case representing pairs as procedures wasn’t mind-boggling enough, consider that, in a language that can manipulate procedures, we can get by without numbers (at least insofar as nonnegative integers are concerned) by implementing 0 and the operation of adding 1 as

``````(define zero (lambda (f) (lambda (x) x)))

(lambda (f) (lambda (x) (f ((n f) x)))))``````

This representation is known as Church numerals, after its inventor, Alonzo Church, the logician who invented the λ calculus.

Define `one` and `two` directly (not in terms of `zero` and `add-1`). (Hint: Use substitution to evaluate `(add-1 zero)`). Give a direct definition of the addition procedure `+` (not in terms of repeated application of `add-1`).

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