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11. Classification (Sets and Propositions)
The list`-.+./t` appended to any classification table`t` will yield a complete classification table, and the function defined below therefore completes a classification table. The function`tab` ensures that a scalar or vector argument is treated as a one-rowed table.

       c=: complete=: (] , (+./ {. ,:)@:-.@:(+./))@:tab
       tab=: ,:^:(0:>.2:-#@$)

       c 0 0 1,:0 1 0
    0 0 1
    0 1 0
    1 0 0

       c 1 0 1,:0 1 0
    1 0 1
    0 1 0

       (c 1 0 1);(c c 1 0 1);(c 0);(c 1)
    +-----+-----+-+-+
    |1 0 1|1 0 1|0|1|
    |0 1 0|0 1 0|1| |
    +-----+-----+-+-+

A function that yields a single boolean list is called a proposition; its result is a one-way classification called a set. The classification can, of course, be completed by the complementary set. For example:

       p1=: 2&<: *. <&5           Set defined by interval
       p1a=: (2:<:]) *. (]<5:)    Alternative definition
       p2=: = <.                  Set of integers
       a=: 2 %~ i. 11
       (],p1,p1a,p2,(p1+.p2),:(p1*.p2)) a
    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
    0   0 0   0 1   1 1   1 1   1 0
    0   0 0   0 1   1 1   1 1   1 0
    1   0 1   0 1   0 1   0 1   0 1
    1   0 1   0 1   1 1   1 1   1 1
    0   0 0   0 1   0 1   0 1   0 0

       list=: 1 : 0               Adverb to list elements of set
    x. # ]
    )

       ((p1 list);(p2 list);((p1*.p2)list)) a
    +-----------------+-----------+-----+
    |2 2.5 3 3.5 4 4.5|0 1 2 3 4 5|2 3 4|
    +-----------------+-----------+-----+

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