Chapter 3Elementary Discrete Mathslides.en.pdf:167-170

Currying & Notations

f(a,b) → f(a)(b); prefix, postfix, infix

Def 4.3 — CurryingDef 4.8 — λ-notationDef 4.6 — Fixity

Concept

Currying transforms an $n$ -ary function into a chain of unary functions.

\text{curry}(f): A_1 \to (A_2 \to (\dots \to (A_n \to B)))

Instead of $f(a, b, c)$ , you write $f(a)(b)(c)$ — apply one argument at a time.

Why? Each partial application can be stored, reused, or passed around.
Adding 3 to a number is a function that waits for the second argument.

λ-notation: $\lambda x \in X. E$ is the function $\{(x, E) \mid x \in X\}$ .
In SML: fn x => E.

Fixity / notation styles:
- Prefix: $f(a, b)$ — function before arguments
- Postfix: $a!$ — function after argument
- Infix: $a + b$ — function between arguments (used for binary ops)
- Circumfix: $(a, b)$ — function wraps argument

Animation — currying

Transcript — click a line to jump8 cues

0.0sCurrying: Cartesian to Cascaded
1.3sCartesian form on the left: (x, y) maps to x + y
2.6sCascaded form on the right: x maps to y maps to x + y
5.6sThe comma flashes amber in the Cartesian form
6.2sThe single arrow flashes amber
6.8sBoth arrows flash amber in the cascaded form
7.9sA double-headed arrow says the two forms are equivalent
9.1sTwo equivalent ways to write a 2-argument function

Practice — score 100% to advance

Multiple choice

What is currying?

\lambda x. x + 1

, what is

x

In SML, what is the expression fn x => x + 1?

Which notation is infix?

In SML, what is add 3 (where add takes two args)?

Loading…