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**13. Compositions (Based on Conjunctions)**

In math, the
symbol` `**°**` `
is commonly used to produce a function defined as the *composition* of two
functions:` f`** ° **`g y `
is defined as` f (g y)` .` `Normally, such composed functions are
only defined to apply to a single scalar argument.

**J** provides compositions effected by five distinct conjunctions,
as well as compositions effected by isolated sequences of verbs:
hooks and forks, and longer trains formed from them.
The five conjunctions are` & &. &: @ `and` @:` ,` `
the conjunctions` @ `and` @: `being related in
the same manner as` & `and` &:` .

The conjunction` & `is closest to the
composition` `**°**` `
used in math, being identical to it when used for two scalar (rank zero)
functions to produce a function to be applied to a single scalar argument.
However, it is also extended in two directions:

1. |
Applied to one verb and one noun it produces
a monadic function illustrated by the cases` 10&^. `
(Base ten logarithm) and` ^&3 `(Cube). |

2. |
Applied to two verbs it produces (in addition to the
monadic case used in math) a dyadic case defined
by:` x f&g y `↔` (g x) f (g y)` .
For example,` x %&! y `is the quotient of
the factorials of` x `and` y` . |

The conjunction` &. `applies only to verbs,
and` f&.g `is equivalent to` f&g `except that
the inverse of` g `is applied to the final result. For example:

3 +&^. 4 3 +&.^. 4
2.48491 12

For scalar arguments the functions` f&:g `
and` f&g `are equivalent, but for more general
arguments,` g `applies to each cell as dictated by its ranks.
In the case of` f&g`,` `the function` f `
then applies to each result produced; in the case of` f&:g `
it applies to the overall result of all of the cells. For example:

(] ; %. ; |:&%. ; |:&:%.) i. 2 2 2
+---+--------+-------+---------+
|0 1|_1.5 0.5|_1.5 1|_1.5 _3.5|
|2 3| 1 0| 0.5 0| 1 3|
| | | | |
|4 5|_3.5 2.5|_3.5 3| 0.5 2.5|
|6 7| 3 _2| 2.5 _2| 0 _2|
+---+--------+-------+---------+

The conjunctions` @ `and` & `
agree in the monadic case, as indicated below for
cells` x `and` y `as dictated by the ranks
of` g` :

` f&g y `↔` f g y`

` f@g y `↔` f g y`

`x f&g y `↔` (g x) f (g y)`

`x f@g y `↔` f (x g y)`

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