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27. Identity Functions and Neutral

The monads 0&+ and 1&* are identity functions, and 0 and 1 are said to be identity elements or neutrals of the dyads + and * respectively. Insertion on an empty list yields the neutral of the dyad inserted. For example:
   +/ i.0         +/''           +/0{. 2 3 5
0              0              0

   */i.0          */''           */0{. 2 3 5
1              1              1
These results are useful in partitioning lists; they ensure that certain obvious relations continue to hold even when one of the partitions is empty. For example:
   +/ a=: 2 3 5 7 11
28

   (+/4{.a)+(+/4}.a)
28

   (+/0{.a)+(+/0}.a)
28

   */a
2310

   (*/4{.a)*(*/4}.a)
2310

   (*/0{.a)*(*/0}.a)
2310
The identity functions and other basic characteristics of functions (such as rank) are given by the adverb b. , as illustrated below:
   ^ b. _1     Inverse
^.

   ^ b. 0      Ranks
_ 0 0

   ^ b. 1      Identity function
$&1@(}.@$)

Exercises

27.1   Predict and test the results of the following expressions: 
*/''
<./''
>./''
>./0 4 4 $ 0
+/ . */ 0 4 4 $ 0
1 2 3 4 +&.^./ 5 6 7 8
27.2   Experiment with the dyad {@; and give the term used to describe it in mathematics.

Answer: Cartesian product
 
27.3   Test the assertion that the monads (%:@~. +/ . * =) and %: are equivalent, and state the utility of the former when applied to a list such as 1 4 1 4 2 that has repeated elements.

Answer: The function %: (which could be a function costly to execute) is applied only to the distinct elements the argument (as selected by the nub function ~.).
 
27.4   Comment on the following experiments before reading the comments on the right:

a=: 2 3 5 [ b=: 1 2 4
a (f=: *:@+) b Square of sum
a (g=: +&*: + +:@*) b   Sum of squares plus double product
a (f=g) b Expression of the identity of the functions
a (f-:g) b f and g in a tautology (whose result is
taut=: f-:g always true; that is, 1).

27.5   A phrase such as f-:g may be a tautology for the dyadic case only, for the monadic case only, or for both. Use the following tautologies as reading and writing exercises, including statements of applicability (Dyad only, etc.):

t1=: >: -: > +. = (Dyad only) The primitive >: is identical to greater than or equal
t2=: <. -: -@>.&- (Both) Lesser-of is neg on greater-of on neg; Floor is neg on ceiling on neg
t3=: <. -: >.&.- Same as t2 but uses under
t4=: *:@>: -: *: + +: + 1: (Monad) Square of a+1 is square of a plus twice a plus 1
t5=: *:@>: -: #.&1 2 1"0 Same as t4 using polynomial
t6=: ^&3@>: -: #.&1 3 3 1"0 Like t5 for cube
bc=: i.@>: ! ] Binomial coefficients
t7=: (>:@]^[) -: (]#.bc@[)"0 Like t6 with k&t7 for kth power
s=: 1&o. Sine
c=: 2&o. Cosine
t8=: s@+-:(s@[*c@])+(c@[*s@])   (Dyad) Addition and Subtraction
t9=: s@--:(s@[*c@])-(c@[*s@]) Formulas for sine
det=: -/ . * Determinant
perm=: +/ . * Permanent
sct=: 1 2&o."0@(,"0) Sine and cosine tables
t10=: s@- -: det@sct Same as t9 but using the determinant of the sin and cos table
t11=: s@+ -: perm@sct Like t8 using the permanent
S=: 5&o. Hyperbolic sine
C=: 6&o. Hyperbolic cosine
SCT=: 5 6&o."0@(,"0) Sinh and Cosh table
t12=: S@+ -: perm@SCT Addition theorem for sinh
SINH=: ^ .: - Odd part of exponential
COSH=: ^ .. - Even part of exponential
t13=: SINH -: S Sinh is odd part of exponential
t14=: COSH -: C Cosh is the even part of exponential
sine=: ^&.j. .: - Sine is the odd part of exponential
t15=: sine -: s under multiplication by 0j1

27.6   Comment on the following expressions before reading the comments on the right:

g=: + > >. Test if sum exceeds maximum
5 g 2 True for positive arguments
5 g _2 _1 0 1 2 but not true in general
f=: *.&(0&<) Test if both arguments exceed 0
theorem=: f <: g The truth value of the result of f does not exceed that of g . This may also be stated as “if f (is true) then g (is true)” or as “f implies g”
5 theorem _2 _1 0 1 2


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