Recall our definitions of true and false.
-> **true** defined to be `\t \f. t`
-> **false** defined to be `\t \f. f`
+> **true** is defined to be `\t \f. t`
+> **false** is defined to be `\t \f. f`
In Racket, these can be defined like this:
<LI>Define an `and` operator.
-<LI>Define an `xor` operator.
-
-If you haven't seen this term before, here's a truth table:
+<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
Recall our definitions of ordered pairs.
-> the pair **(**x**,**y**)** is defined as `\f. f x y`
+> the pair **(**x**,**y**)** is defined to be `\f. f x y`
To extract the first element of a pair p, you write:
(p get-first) ; will evaluate to 10
(p get-second) ; will evaluate to 20
-If you're bothered by having the pair to the left and the function that
+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 elemens with that
-function. In other words, the pair is also a function. (Of course, in the
-untyped lambda calculus, absolutely *everything* is a function: functors,
-arguments, abstracts, redexes, values---everything.)
+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:
(p get-first)
-However, the latter is still what's going on under the hood.
+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:
+<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.
+Write out the definition of `swap` in Racket.
<LI>Define a `dup` function that duplicates its argument to form a pair