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{{Main|Comparison function}}
{{Main|Comparison function}}


{{Built-in|Less than or Equal to|<nowiki>≤</nowiki>}} is a [[comparison function]] which tests whether the left [[argument]] is [[Tolerant comparison|tolerantly]] less than or equal to the right argument, returning 1 if this is the case and 0 otherwise. It is the [[Not|negation]] of [[Less than]] (<source lang=apl inline><</source>), and in fact was called '''Not greater''' in [[APL\360]].
{{Built-in|Less than or Equal to|<nowiki>≤</nowiki>}} is a [[comparison function]] which tests whether the left [[argument]] is [[Tolerant comparison|tolerantly]] less than or equal to the right argument, returning 1 if this is the case and 0 otherwise. It is the [[Not|negation]] of [[Less than]] (<syntaxhighlight lang=apl inline><</syntaxhighlight>), and in fact was called '''Not greater''' in [[APL\360]].


== Boolean function ==
== Boolean function ==
{{Main|Boolean function}}
When the arguments to Less than or Equal to are [[Boolean]], it is the [[wikipedia:Material conditional|material implication]] [[Boolean function]], also known as the [[wikipedia:IMPLY gate|IMPLY gate]]:
When used as a [[Boolean function]] (that is, with argument consisting entirely of 0s and 1s), it represents [[wikipedia:Material conditional|material implication]], also known as the [[wikipedia:IMPLY gate|IMPLY gate]]:
::{|class=wikitable
{|class=wikitable
!<syntaxhighlight lang=apl inline>≤</syntaxhighlight>!!<syntaxhighlight lang=apl inline>0</syntaxhighlight>!!<syntaxhighlight lang=apl inline>1</syntaxhighlight>
!<source lang=apl inline>≤</source>!!<source lang=apl inline>0</source>!!<source lang=apl inline>1</source>
|-
|-
!<source lang=apl inline>0</source>
!<syntaxhighlight lang=apl inline>0</syntaxhighlight>
|<source lang=apl inline>1</source>||<source lang=apl inline>1</source>
|<syntaxhighlight lang=apl inline>1</syntaxhighlight>||<syntaxhighlight lang=apl inline>1</syntaxhighlight>
|-
|-
!<source lang=apl inline>1</source>
!<syntaxhighlight lang=apl inline>1</syntaxhighlight>
|<source lang=apl inline>0</source>||<source lang=apl inline>1</source>
|<syntaxhighlight lang=apl inline>0</syntaxhighlight>||<syntaxhighlight lang=apl inline>1</syntaxhighlight>
|}
|}
In the context of logic, it can be read as ''implies''.  
In the context of logic, it can be read as ''implies''.
 
Less than or Equal to [[Scan]] is an occasionally used pattern related to [[Less than]] Scan. While <syntaxhighlight lang=apl inline><\</syntaxhighlight> changes all 1s after the first to 0, <syntaxhighlight lang=apl inline>≤\</syntaxhighlight> changes all 0s after the first to 1. Thus <syntaxhighlight lang=apl inline>≤\A</syntaxhighlight> {{←→}} <syntaxhighlight lang=apl inline>~<\~A</syntaxhighlight>.
<syntaxhighlight lang=apl>
      ≤\ 1 0 1 0 0 0 1 0
1 0 1 1 1 1 1 1
</syntaxhighlight>
<syntaxhighlight lang=apl inline>≤\</syntaxhighlight> appears in the [[FinnAPL idiom library]] as entry 350, "Not first zero".
 
== External links ==
== External links ==


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


* [http://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Less%20Or%20Equal.htm Dyalog]
* [https://help.dyalog.com/latest/index.htm#Language/Primitive%20Functions/Less%20Or%20Equal.htm Dyalog]
* [http://microapl.com/apl_help/ch_020_020_300.htm APLX]
* [http://microapl.com/apl_help/ch_020_020_300.htm APLX]
* J [http://www.jsoftware.com/help/dictionary/d012.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/ltco#dyadic NuVoc]
* J [http://www.jsoftware.com/help/dictionary/d012.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/ltco#dyadic NuVoc]
* [https://mlochbaum.github.io/BQN/doc/arithmetic.html#comparisons BQN]


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

Latest revision as of 22:20, 10 September 2022

Main article: Comparison function
≤

Less than or Equal to (≤) is a comparison function which tests whether the left argument is tolerantly less than or equal to the right argument, returning 1 if this is the case and 0 otherwise. It is the negation of Less than (<), and in fact was called Not greater in APL\360.

Boolean function

When the arguments to Less than or Equal to are Boolean, it is the material implication Boolean function, also known as the IMPLY gate:

≤ 0 1
0 1 1
1 0 1

In the context of logic, it can be read as implies.

Less than or Equal to Scan is an occasionally used pattern related to Less than Scan. While <\ changes all 1s after the first to 0, ≤\ changes all 0s after the first to 1. Thus ≤\A ~<\~A. 

      ≤\ 1 0 1 0 0 0 1 0
1 0 1 1 1 1 1 1

≤\ appears in the FinnAPL idiom library as entry 350, "Not first zero".

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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