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{{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]] <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 ==
== Extension ==
Some dialects allow Without to work on [[major cell]]s:
Some dialects allow Without to work on [[major cell]]s:
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihojBB0dFTnetQ3FSgYURf5/z8A Try it online!]<source lang=apl>
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihojBB0dFTnetQ3FSgYURf5/z8A Try it online!]<syntaxhighlight lang=apl>
       X←4 2⍴'AABBCCCC'
       X←4 2⍴'AABBCCCC'
       Y←3 2⍴'AABBAA'
       Y←3 2⍴'AABBAA'
Line 23: Line 23:
CC
CC
CC
CC
</source>{{Works in|[[Extended Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Extended Dyalog APL]]}}
Others can easily define such a function:
Others can easily define such a function:
[https://tio.run/##SyzI0U2pTMzJT///P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihojBB0dFTnCs8sycgvLQFKVD/q3fWoZ/@j3hV1Go96Nz/qXAQU0HzU0QUSB/J7t9ZyPeqbClQZoQDVpRD5/z8A Try it online!]<source lang=apl>
[https://tio.run/##SyzI0U2pTMzJT///P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihojBB0dFTnCs8sycgvLQFKVD/q3fWoZ/@j3hV1Go96Nz/qXAQU0HzU0QUSB/J7t9ZyPeqbClQZoQDVpRD5/z8A Try it online!]<syntaxhighlight lang=apl>
       X←4 2⍴'AABBCCCC'
       X←4 2⍴'AABBCCCC'
       Y←3 2⍴'AABBAA'
       Y←3 2⍴'AABBAA'
Line 32: Line 32:
CC
CC
CC
CC
</source>{{Works in|[[Dyalog APL]]}}
</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
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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