The Wolfram Language includes many powerful operations for working with lists. It is often desirable to map a function onto each individual element in a list. While listable functions do this by default, you can use Map to do this with non-listable functions.

First set up a list of the integers from 1 to 5:

You can map a function over every element of the list using Map; this example uses an undefined function f:

You can use /@ as a shorthand for Map (this is the same command as in the previous example):

Most mathematical functions have the Listable property, meaning they automatically map over lists:

If the function is not Listable, you can use Map, instead. Set up a list of five 2×2 matrices:

Use Map to map MatrixForm over the list to see each of them in mathematical notation:

Now use Map to calculate the eigenvalues of each of the matrices in the list:

Map does not just operate on lists. It can be used for any expression:

Apply is another functional programming operation. It replaces the head of an expression.

You can see how this works using two undefined functions, f and g:

Apply has @@ for a shorthand notation (this is the same command as the previous example):

Common expressions are shown in StandardForm in the Wolfram Language, but their underlying FullForm shows how Apply can be used:

For example, this changes a sum into a product:

Apply is useful when you want to turn the elements in a list into function arguments.

Create a list of five ordered pairs {a,b}:

Mod finds the remainder when dividing the first number of an ordered pair by the second:

To apply Mod to all of the pairs, you need to work at level 1 of the list (specified by the {1}):

You can use @@@ as a shorthand to apply at level 1:

This is another way to do the same thing using a pure function with Function:

This uses the short form of Function: