a set of ordered pairs. This set is called the **graph** of the
function. If the ordered pair `(a, b)` is a member of the graph of `f`,
that means that `f` maps the argument `a` to the value `b`. In
-symbols, `f: a` ↦ `b`, or `f (a) == b`.
+symbols, `f: a` ↦ `b`, or `f (a) == b`.
In order to count as a *function* (rather
than as merely a more general *relation*), we require that the graph not contain two
elements is all that it takes to simulate the behavior of a general
word processing program. That means that if we had a big enough
lambda term, it could take a representation of *Emma* as input and
-produce *Hamlet* as a result.
+produce *Hamlet* as a result.
Some of these functions are so useful that we'll give them special
names. In particular, we'll call the identity function `(\x x)`
(\z z)
-yet when applied to any argument M, all of these will always return M. So they
+yet when applied to any argument `M`, all of these will always return `M`. So they
have the same extension. It's also true, though you may not yet be in a
position to see, that no other function can differentiate between them when
they're supplied as an argument to it. However, these expressions are all
## The analogy with `let` ##
In our basic functional programming language, we used `let`
-expressions to assign values to variables. For instance,
+expressions to assign values to variables. For instance,
let x match 2
- in (x, x)
+ in (x, x)
-evaluates to the ordered pair (2, 2). It may be helpful to think of
+evaluates to the ordered pair `(2, 2)`. It may be helpful to think of
a redex in the lambda calculus as a particular sort of `let`
construction.
is analogous to
- let x match ARG
+ let x match ARG
in BODY
This analogy should be treated with caution. For one thing, our `letrec`