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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]] <source 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" <source 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>
<source lang=apl>
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│THIS│THAT│
│THIS│THAT│
└────┴────┘
└────┴────┘
</source>
</syntaxhighlight>


== Extension ==
== Extension ==
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</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!]<source lang=apl>
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</source>{{Works in|[[Dyalog APL]]}}
</syntaxhighlight>{{Works in|[[Dyalog APL]]}}
== External Links ==
== External Links ==


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