Union: Difference between revisions

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== Examples ==
== Examples ==


Both arguments of Intersection is usually restricted to [[vector|vectors]]. Unlike sets in the mathematical sense, duplicate elements are allowed in both sides, and Union is usually implemented as "left argument unchanged, plus the elements of right argument not in the left argument" <source lang=apl inline>X,(~Y∊X)/Y</syntaxhighlight>. This preserves the order and multiplicity in both arguments, the left one taking precedence. Both arguments can be [[nested array|nested arrays]].
Both arguments of Intersection is usually restricted to [[vector|vectors]]. Unlike sets in the mathematical sense, duplicate elements are allowed in both sides, and Union is usually implemented as "left argument unchanged, plus the elements of right argument not in the left argument" <syntaxhighlight lang=apl inline>X,(~Y∊X)/Y</syntaxhighlight>. This preserves the order and multiplicity in both arguments, the left one taking precedence. Both arguments can be [[nested array|nested arrays]].


<source lang=apl>
<syntaxhighlight lang=apl>
       'WASH' ∪ 'SHOUT'
       'WASH' ∪ 'SHOUT'
WASHOUT
WASHOUT
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== Extension ==
== Extension ==
Some dialects allow Union to work on [[major cell]]s:
Some dialects allow Union to work on [[major cell]]s:
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihoDBN0cXF0VOd61DcVKBjxqGNV5P//AA Try it online!]<source lang=apl>
[https://tio.run/##SyzI0U2pTMzJT9dNrShJzUtJTfn/P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihoDBN0cXF0VOd61DcVKBjxqGNV5P//AA Try it online!]<syntaxhighlight lang=apl>
       X←4 2⍴'AABBCCCC'
       X←4 2⍴'AABBCCCC'
       Y←3 2⍴'AADDAA'
       Y←3 2⍴'AADDAA'
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</syntaxhighlight>{{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@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihoDBN0cXF0VOcKzcvMzwMKVz/q3fWod9Wj3q2PevY/6l2h8ahzEVBI0wYsvhkoXsv1qG8qUGWEAliPQuT//wA Try it online!]<source lang=apl>
[https://tio.run/##SyzI0U2pTMzJT///P@JR2wQTBaNHvVvUHR2dnJyBQJ0rEihoDBN0cXF0VOcKzcvMzwMKVz/q3fWod9Wj3q2PevY/6l2h8ahzEVBI0wYsvhkoXsv1qG8qUGWEAliPQuT//wA Try it online!]<syntaxhighlight lang=apl>
       X←4 2⍴'AABBCCCC'
       X←4 2⍴'AABBCCCC'
       Y←3 2⍴'AADDAA'
       Y←3 2⍴'AADDAA'
       Intersection←{⍺⌿⍨(≢⍵)≥⍵⍳⍺}
       Union←{⍺⍪⍵⌿⍨(≢⍺)<⍺⍳⍵}
       X Intersection Y
       X Union Y
AA
AA
BB
BB

Latest revision as of 17:25, 6 January 2024

∪

Union (∪) is a dyadic set function which computes the set union of the two vector arguments.

Examples

Both arguments of Intersection is usually restricted to vectors. Unlike sets in the mathematical sense, duplicate elements are allowed in both sides, and Union is usually implemented as "left argument unchanged, plus the elements of right argument not in the left argument" X,(~Y∊X)/Y. This preserves the order and multiplicity in both arguments, the left one taking precedence. Both arguments can be nested arrays. 

      'WASH' ∪ 'SHOUT'
WASHOUT
      1 2 3 1 2 3 1 2 3∪5 4 3 2 5 4 3 2
1 2 3 1 2 3 1 2 3 5 4 5 4
      'THIS' 'THAT' 'THE' 'OTHER'∪'OTHER' 'THAN' 'THIS' 'AND' 'THAT'
┌────┬────┬───┬─────┬────┬───┐
│THIS│THAT│THE│OTHER│THAN│AND│
└────┴────┴───┴─────┴────┴───┘

Extension

Some dialects allow Union to work on major cells:

Try it online!

      X←4 2⍴'AABBCCCC'
      Y←3 2⍴'AADDAA'
      X∩Y
AA
BB
CC
CC
DD
Works in: Extended Dyalog APL

Others can easily define such a function:

Try it online!

      X←4 2⍴'AABBCCCC'
      Y←3 2⍴'AADDAA'
      Union←{⍺⍪⍵⌿⍨(≢⍺)<⍺⍳⍵}
      X Union Y
AA
BB
CC
CC
DD
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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