Pure Functions and Map

A pure function is a function without a name — written right where you need it.

(#^2 &)[5]
(#1 + #2 &)[3, 4]
Function[x, x^2][5]
(x |-> x^2)[5]

25
7
25
25

• In #^2 &, # is the argument and & marks the end of the function. #1, #2, ... are the first, second, ... arguments.
• Function[x, body] and x |-> body are the same thing with a named argument — clearer when the body is long.

Map: /@

Map[f, list], written f /@ list, applies f to every element:

f /@ {a, b, c}
#^2 & /@ {1, 2, 3}
StringLength /@ {"kiwi", "apple", "banana"}

{f[a], f[b], f[c]}
{1, 4, 9}
{4, 5, 6}

(For arithmetic, listability does the same: {1, 2, 3}^2. Map works with any function.)

Select with a pure function

Select[Range[20], # > 15 &]
Select[{"apple", "kiwi", "banana"}, StringLength[#] > 4 &]
SortBy[{"banana", "kiwi", "apple"}, StringLength]

{16, 17, 18, 19, 20}
{apple, banana}
{kiwi, apple, banana}

Apply: @@ and @@@

Apply[f, expr], written f @@ expr, replaces the head:

f @@ {a, b, c}
Plus @@ {1, 2, 3}
Times @@ {2, 3, 4}
f @@@ {{1, 2}, {3, 4}}

f[a, b, c]
6
24
{f[1, 2], f[3, 4]}

• Plus @@ {1, 2, 3} turns List[1, 2, 3] into Plus[1, 2, 3].
• @@@ applies at level 1 — to each inner list. Useful for lists of pairs: (#1 * #2 &) @@@ {{2, 3}, {4, 5}} gives {6, 20}.

Summary of the shorthand

Common mistakes

• Forgetting the & — #^2 /@ {1, 2} doesn't define a function.
• Precedence: & binds loosely, so Select[list, # > 2 && # < 5 &] works, but when combining with other operators, parentheses — (# + 1 &) /@ list — keep things clear.

Practice

Given {{"Ada", 36}, {"Alan", 41}}, produce the strings "Ada is 36" and "Alan is 41".

(#1 <> " is " <> ToString[#2] &) @@@ {{"Ada", 36}, {"Alan", 41}}

{Ada is 36, Alan is 41}