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{{Built-in|Without|~}}, '''Excluding''', or '''Less''', is a [[dyadic]] [[set function]] which computes the [[wikipedia:set difference|set difference]] of the two [[vector]] [[argument|arguments]]. It shares the [[glyph]] <source lang=apl inline>~</source> with the monadic function [[Not]].
{{Built-in|Without|~}}, '''Set Difference''', '''Excluding''', or '''Less''', is a [[dyadic]] [[set function]] which computes the [[wikipedia:set difference|set difference]] of the two [[vector]] [[argument|arguments]]. It shares the [[glyph]] <syntaxhighlight lang=apl inline>~</syntaxhighlight> with the monadic function [[Not]].


== Examples ==
== Examples ==


Both arguments of Without is usually restricted to [[vector|vectors]]. Unlike sets in the mathematical sense, duplicate elements are allowed in both sides, and Without is usually implemented as "left argument filtered by the absence in the right argument" <source lang=apl inline>(~X∊Y)/X</source>. This preserves the order and multiplicity in the left argument. Both arguments can be [[nested array|nested arrays]].
Both arguments of Without is usually restricted to [[vector|vectors]]. Unlike sets in the mathematical sense, duplicate elements are allowed in both sides, and Without is usually implemented as "left argument filtered by the absence in the right argument" <syntaxhighlight lang=apl inline>(~X∊Y)/X</syntaxhighlight>. This preserves the order and multiplicity in the left argument. Both arguments can be [[nested array|nested arrays]].


<source lang=apl>
<syntaxhighlight lang=apl>
       'HELLO'~'GOODBYE'
       'HELLO'~'GOODBYE'
HLL
HLL
Line 13: Line 13:
│THIS│THAT│
│THIS│THAT│
└────┴────┘
└────┴────┘
</source>
</syntaxhighlight>


== Extension ==
Some dialects allow Without to work on [[major cell]]s:
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihojBB0dFTnetQ3FSgYURf5/z8A Try it online!]<syntaxhighlight lang=apl>
      X←4 2⍴'AABBCCCC'
      Y←3 2⍴'AABBAA'
      X~Y
CC
CC
</syntaxhighlight>{{Works in|[[Extended Dyalog APL]]}}
Others can easily define such a function:
[https://tio.run/##SyzI0U2pTMzJT///P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihojBB0dFTnCs8sycgvLQFKVD/q3fWoZ/@j3hV1Go96Nz/qXAQU0HzU0QUSB/J7t9ZyPeqbClQZoQDVpRD5/z8A Try it online!]<syntaxhighlight lang=apl>
      X←4 2⍴'AABBCCCC'
      Y←3 2⍴'AABBAA'
      Without←{⍺⌿⍨~(⍳≢⍺)∊⍺⍳⍵}
      X Without Y
CC
CC
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}
== External Links ==
== External Links ==


=== Documentation ===
=== Documentation ===


* [http://help.dyalog.com/latest/#Language/Primitive%20Functions/Excluding.htm#Excluding Dyalog]
* [https://help.dyalog.com/latest/#Language/Primitive%20Functions/Excluding.htm#Excluding Dyalog]
* [http://microapl.com/apl_help/ch_020_020_410.htm APLX]
* [http://microapl.com/apl_help/ch_020_020_410.htm APLX]
* J [https://www.jsoftware.com/help/dictionary/d121.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/minusdot#dyadic NuVoc]
* J [https://www.jsoftware.com/help/dictionary/d121.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/minusdot#dyadic NuVoc]


{{APL built-ins}}[[Category:Primitive functions]][[Category:Set functions]]
{{APL built-ins}}[[Category:Primitive functions]][[Category:Set functions]]

Latest revision as of 10:58, 11 September 2022

~

Without (~), Set Difference, Excluding, or Less, is a dyadic set function which computes the set difference of the two vector arguments. It shares the glyph ~ with the monadic function Not.

Examples

Both arguments of Without is usually restricted to vectors. Unlike sets in the mathematical sense, duplicate elements are allowed in both sides, and Without is usually implemented as "left argument filtered by the absence in the right argument" (~X∊Y)/X. This preserves the order and multiplicity in the left argument. Both arguments can be nested arrays. 

      'HELLO'~'GOODBYE'
HLL
      A←'THIS' 'AND' 'THAT'
      A~'TH' 'AND'
┌────┬────┐
│THIS│THAT│
└────┴────┘

Extension

Some dialects allow Without to work on major cells:

Try it online!

      X←4 2⍴'AABBCCCC'
      Y←3 2⍴'AABBAA'
      X~Y
CC
CC
Works in: Extended Dyalog APL

Others can easily define such a function:

Try it online!

      X←4 2⍴'AABBCCCC'
      Y←3 2⍴'AABBAA'
      Without←{⍺⌿⍨~(⍳≢⍺)∊⍺⍳⍵}
      X Without Y
CC
CC
Works in: Dyalog APL

External Links

Documentation


APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
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Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
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Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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