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 <source lang=apl inline>~</syntaxhighlight> 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" <source lang=apl inline>(~X∊Y)/X</syntaxhighlight>. This preserves the order and multiplicity in the left argument. Both arguments can be nested arrays.
<source lang=apl>
'HELLO'~'GOODBYE'
HLL
A←'THIS' 'AND' 'THAT'
A~'TH' 'AND'
┌────┬────┐
│THIS│THAT│
└────┴────┘
</syntaxhighlight>
Extension
Some dialects allow Without to work on major cells:
Try it online!<source lang=apl>
X←4 2⍴'AABBCCCC'
Y←3 2⍴'AABBAA'
X~Y
CC
CC
</syntaxhighlight>
Others can easily define such a function:
Try it online!<source lang=apl>
X←4 2⍴'AABBCCCC'
Y←3 2⍴'AABBAA'
Without←{⍺⌿⍨~(⍳≢⍺)∊⍺⍳⍵}
X Without Y
CC
CC
</syntaxhighlight>
External Links
Documentation
| APL built-ins [edit]
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| Primitives (Timeline) |
Functions
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| Scalar
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| Monadic
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Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
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| Dyadic
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Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
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| Non-Scalar
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| Structural
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Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
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| Selection
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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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| Selector
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Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
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| Computational
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Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
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| Operators |
Monadic
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Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
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| Dyadic
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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)
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| Quad names
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Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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