--- /dev/null
+Reduction
+---------
+
+Find "normal forms" for the following---that is, reduce them until no more reductions are possible. We'll write <code>λx</code> as `\x`.
+
+1. `(\x \y. y x) z`
+2. `(\x (x x)) z`
+3. `(\x (\x x)) z`
+4. `(\x (\z x)) z`
+5. `(\x (x (\y y))) (\z (z z))`
+6. `(\x (x x)) (\x (x x))`
+7. `(\x (x x x)) (\x (x x x))`
+
+
+Booleans
+--------
+
+Recall our definitions of true and false.
+
+> **true** is defined to be `\t \f. t`
+> **false** is defined to be `\t \f. f`
+
+In Racket, these can be defined like this:
+
+ (define true (lambda (t) (lambda (f) t)))
+ (define false (lambda (t) (lambda (f) f)))
+
+<OL start=8>
+<LI>Define a `neg` operator that negates `true` and `false`.
+
+Expected behavior:
+
+ (((neg true) 10) 20)
+
+evaluates to 20, and
+
+ (((neg false) 10) 20)
+
+evaluates to 10.
+
+<LI>Define an `and` operator.
+
+<LI>Define an `xor` operator. If you haven't seen this term before, here's a truth table:
+
+ true xor true = false
+ true xor false = true
+ false xor true = true
+ false xor false = false
+
+
+<LI>Inspired by our definition of boolean values, propose a data structure
+capable of representing one of the two values `black` or `white`.
+If we have
+one of those values, call it a "black-or-white value", we should be able to
+write:
+
+ the-value if-black if-white
+
+(where `if-black` and `if-white` are anything), and get back one of `if-black` or
+`if-white`, depending on which of the black-or-white values we started with. Give
+a definition for each of `black` and `white`. (Do it in both lambda calculus
+and also in Racket.)
+
+<LI>Now propose a data structure capable of representing one of the three values
+`red` `green` or `blue`, based on the same model. (Do it in both lambda
+calculus and also in Racket.)
+</OL>
+
+
+
+Pairs
+-----
+
+Recall our definitions of ordered pairs.
+
+> the pair **(**x**,**y**)** is defined to be `\f. f x y`
+
+To extract the first element of a pair p, you write:
+
+ p (\fst \snd. fst)
+
+Here are some definitions in Racket:
+
+ (define make-pair (lambda (fst) (lambda (snd) (lambda (f) ((f fst) snd)))))
+ (define get-first (lambda (fst) (lambda (snd) fst)))
+ (define get-second (lambda (fst) (lambda (snd) snd)))
+
+Now we can write:
+
+ (define p ((make-pair 10) 20))
+ (p get-first) ; will evaluate to 10
+ (p get-second) ; will evaluate to 20
+
+If you're puzzled by having the pair to the left and the function that
+operates on it come second, think about why it's being done this way: the pair
+is a package that takes a function for operating on its elements *as an
+argument*, and returns *the result of* operating on its elements with that
+function. In other words, the pair is a higher-order function. (Consider the similarities between this definition of a pair and a generalized quantifier.)
+
+If you like, you can disguise what's going on like this:
+
+ (define lifted-get-first (lambda (p) (p get-first)))
+ (define lifted-get-second (lambda (p) (p get-second)))
+
+Now you can write:
+
+ (lifted-get-first p)
+
+instead of:
+
+ (p get-first)
+
+However, the latter is still what's going on under the hood. (Remark: `(lifted-f ((make-pair 10) 20))` stands to `(((make-pair 10) 20) f)` as `(((make-pair 10) 20) f)` stands to `((f 10) 20)`.)
+
+
+<OL start=13>
+<LI>Define a `swap` function that reverses the elements of a pair. Expected behavior:
+
+ (define p ((make-pair 10) 20))
+ ((p swap) get-first) ; evaluates to 20
+ ((p swap) get-second) ; evaluates to 10
+
+Write out the definition of `swap` in Racket.
+
+
+<LI>Define a `dup` function that duplicates its argument to form a pair
+whose elements are the same.
+Expected behavior:
+
+ ((dup 10) get-first) ; evaluates to 10
+ ((dup 10) get-second) ; evaluates to 10
+
+<LI>Define a `sixteen` function that makes
+sixteen copies of its argument (and stores them in a data structure of
+your choice).
+
+<LI>Inspired by our definition of ordered pairs, propose a data structure capable of representing ordered triples. That is,
+
+ (((make-triple M) N) P)
+
+should return an object that behaves in a reasonable way to serve as a triple. In addition to defining the `make-triple` function, you have to show how to extract elements of your triple. Write a `get-first-of-triple` function, that does for triples what `get-first` does for pairs. Also write `get-second-of-triple` and `get-third-of-triple` functions.
+
+<LI>Write a function `second-plus-third` that when given to your triple, returns the result of adding the second and third members of the triple.
+
+You can help yourself to the following definition:
+
+ (define add (lambda (x) (lambda (y) (+ x y))))
+
+<!-- Write a function that reverses the order of the elements in a list. [Only attempt this problem if you're feeling frisky, it's super hard unless you have lots of experience programming.] -->
+
+</OL>
+