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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]] (<source lang=apl inline><</syntaxhighlight>), and in fact was called '''Not greater''' in [[APL\360]].


== Boolean function ==
== 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 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]]:
::{|class=wikitable
::{|class=wikitable
!<source lang=apl inline>≤</source>!!<source lang=apl inline>0</source>!!<source lang=apl inline>1</source>
!<source lang=apl inline>≤</syntaxhighlight>!!<source lang=apl inline>0</syntaxhighlight>!!<source lang=apl inline>1</syntaxhighlight>
|-
|-
!<source lang=apl inline>0</source>
!<source lang=apl inline>0</syntaxhighlight>
|<source lang=apl inline>1</source>||<source lang=apl inline>1</source>
|<source lang=apl inline>1</syntaxhighlight>||<source lang=apl inline>1</syntaxhighlight>
|-
|-
!<source lang=apl inline>1</source>
!<source lang=apl inline>1</syntaxhighlight>
|<source lang=apl inline>0</source>||<source lang=apl inline>1</source>
|<source lang=apl inline>0</syntaxhighlight>||<source 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 <source lang=apl inline><\</source> changes all 1s after the first to 0, <source lang=apl inline>≤\</source> changes all 0s after the first to 1. Thus <source lang=apl inline>≤\A</source> {{←→}} <source lang=apl inline>~<\~A</source>.
Less than or Equal to [[Scan]] is an occasionally used pattern related to [[Less than]] Scan. While <source lang=apl inline><\</syntaxhighlight> changes all 1s after the first to 0, <source lang=apl inline>≤\</syntaxhighlight> changes all 0s after the first to 1. Thus <source lang=apl inline>≤\A</syntaxhighlight> {{←→}} <source lang=apl inline>~<\~A</syntaxhighlight>.
<source lang=apl>
<source lang=apl>
       ≤\ 1 0 1 0 0 0 1 0
       ≤\ 1 0 1 0 0 0 1 0
1 0 1 1 1 1 1 1
1 0 1 1 1 1 1 1
</source>
</syntaxhighlight>
<source lang=apl inline>≤\</source> appears in the [[FinnAPL idiom library]] as entry 350, "Not first zero".
<source lang=apl inline>≤\</syntaxhighlight> appears in the [[FinnAPL idiom library]] as entry 350, "Not first zero".


== External links ==
== External links ==

Revision as of 22:15, 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 (<source lang=apl inline><</syntaxhighlight>), 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:

<source lang=apl inline>≤</syntaxhighlight> <source lang=apl inline>0</syntaxhighlight> <source lang=apl inline>1</syntaxhighlight>
<source lang=apl inline>0</syntaxhighlight> <source lang=apl inline>1</syntaxhighlight> <source lang=apl inline>1</syntaxhighlight>
<source lang=apl inline>1</syntaxhighlight> <source lang=apl inline>0</syntaxhighlight> <source lang=apl inline>1</syntaxhighlight>

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 <source lang=apl inline><\</syntaxhighlight> changes all 1s after the first to 0, <source lang=apl inline>≤\</syntaxhighlight> changes all 0s after the first to 1. Thus <source lang=apl inline>≤\A</syntaxhighlight> <source lang=apl inline>~<\~A</syntaxhighlight>. <source lang=apl> 

     ≤\ 1 0 1 0 0 0 1 0

1 0 1 1 1 1 1 1 </syntaxhighlight> <source lang=apl inline>≤\</syntaxhighlight> 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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