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Revision as of 01:55, 24 July 2022

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→‎Pairing monadic and dyadic meanings
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<source lang=apl inline>¯</source>, <source lang=apl inline><</source>, <source lang=apl inline>≤</source>, <source lang=apl inline>=</source>, <source lang=apl inline>≥</source>, <source lang=apl inline>></source>, and <source lang=apl inline>≠</source> form a block. The number line 1–9 (because 0 on the far right) is split into two [[equal]] halves by <kbd>5</kbd> so that gives <source lang=apl inline>=</source>. <kbd>4</kbd> and <kbd>6</kbd> are slightly less and more, respectively, so they give <source lang=apl inline>≤</source> and <source lang=apl inline>≥</source>. <kbd>3</kbd> and <kbd>7</kbd> are much less and more, respectively, so they give <source lang=apl inline><</source> and <source lang=apl inline>></source>. Finally, <kbd>2</kbd> is so much less that it is negative, giving the negative sign <source lang=apl inline>¯</source>, and <kbd>8</kbd> is so much greater that it is completely [[not equal|unequal]], <source lang=apl inline>≠</source>.
<source lang=apl inline>¯</source>, <source lang=apl inline><</source>, <source lang=apl inline>≤</source>, <source lang=apl inline>=</source>, <source lang=apl inline>≥</source>, <source lang=apl inline>></source>, and <source lang=apl inline>≠</source> form a block. The number line 1–9 (because 0 on the far right) is split into two [[equal]] halves by <kbd>5</kbd> so that gives <source lang=apl inline>=</source>. <kbd>4</kbd> and <kbd>6</kbd> are slightly less and more, respectively, so they give <source lang=apl inline>≤</source> and <source lang=apl inline>≥</source>. <kbd>3</kbd> and <kbd>7</kbd> are much less and more, respectively, so they give <source lang=apl inline><</source> and <source lang=apl inline>></source>. Finally, <kbd>2</kbd> is so much less that it is negative, giving the negative sign <source lang=apl inline>¯</source>, and <kbd>8</kbd> is so much greater that it is completely [[not equal|unequal]], <source lang=apl inline>≠</source>.
== Pairing monadic and dyadic meanings ==
Meaning pairings fall into three groups as follows.
=== Monadic form is like dyadic form with a default left argument ===
{| class=wikitable
! Glyph
! Default left argument
|-
| <source lang=apl inline>+</source>
| <source lang=apl inline>0J¯2×11○x</source> i.e. <math>-2\!\cdot\!\Im(x)</math>
|-
| <source lang=apl inline>-</source>
| 0
|-
| <source lang=apl inline>×</source>
| <source lang=apl inline>÷(|x)+x=0</source> i.e. <math>\frac1{|x|+[x=0]}</math>
|-
| <source lang=apl inline>÷</source>
| 1
|-
| <source lang=apl inline>*</source>
| <source lang=apl inline>*1</source> i.e. <math>e</math>
|-
| <source lang=apl inline>⍟</source>
| <source lang=apl inline>*1</source> i.e. <math>e</math>
|-
| <source lang=apl inline>,</source>
| <source lang=apl inline>⍬</source> (for scalars and vectors only)
|-
| <source lang=apl inline>?</source>
| 1 (almost; monadic form gives a scalar, dyadic form gives a vector)
|-
| <source lang=apl inline>⍒</source>
| <source lang=apl inline>⎕UCS 0,⍳111411</source>
|-
| <source lang=apl inline>⍋</source>
| <source lang=apl inline>⎕UCS 0,⍳111411</source>
|-
| <source lang=apl inline>⍉</source>
| <source lang=apl inline>⌽⍳≢⍴x</source>
|-
| <source lang=apl inline>⍎</source>
| <source lang=apl inline>⎕THIS</source>
|-
| <source lang=apl inline>⌹</source>
| <source lang=apl inline>∘.=⍨⍳≢x</source> i.e. <math>I_n</math>
|-
| <source lang=apl inline>⊂</source>
| <source lang=apl inline>,1</source> (almost; monadic form gives a scalar, dyadic form gives a vector)
|-
| <source lang=apl inline>⊃</source>
| <source lang=apl inline>⊂(≢⍴x)⍴1</source>
|-
| <source lang=apl inline>∪</source>
| <source lang=apl inline>∪x</source>
|-
| <source lang=apl inline>⌷</source>
| <source lang=apl inline>⊂⍳≢⌷x</source>
|-
| <source lang=apl inline>⊣</source>
| <source lang=apl inline>x</source>
|-
| <source lang=apl inline>⊢</source>
| <source lang=apl inline>x</source>
|}
=== Both forms are good fit for the glyph ===
{| class=wikitable
! Glyph
! Explanation
|-
| <source lang=apl inline>≡</source>
| indicates (3) layers of depth for the monadic form
|-
| <source lang=apl inline>≢</source>
| indicates a counting rod marking for the monadic form
|-
| <source lang=apl inline>↑</source>
| take elements and increase rank
|-
| <source lang=apl inline>↓</source>
| drop elements and decrease rank
|-
| <source lang=apl inline>∊</source>
| &quot;'''e'''lement of&quot; and &quot;'''e'''nlist&quot;
|-
| <source lang=apl inline>⊆</source>
| &quot;enclose at positions where condition is met&quot; and &quot;enclose on condition of being simple&quot;
|-
| <source lang=apl inline>⍸</source>
| &quot;'''i'''ndices where elements fit into intervals&quot; and &quot;'''i'''ndices where true&quot;
|-
| <source lang=apl inline>⍪</source>
| <source lang=apl inline>,[1]x</source> (for non-scalars) and <source lang=apl inline>,⍤¯1</source> (for higher-rank arrays)
|}
=== The two forms are conceptually related ===
{| class=wikitable
! Glyph
! Relationship
|-
| <source lang=apl inline>⌈</source>
| next higher integer or highest argument
|-
| <source lang=apl inline>⌊</source>
| next lower integer or lowest argument
|-
| <source lang=apl inline>|</source>
| both forms can be called [[wikipedia:modulo|modulo]] and use the symbol: <math>|x|</math> and <math>a|b</math>
|-
| <source lang=apl inline>!</source>
| factorial is prominent in the formula for binomial
|-
| <source lang=apl inline>○</source>
| both the trigonometric functions and <math>\pi</math> relate to the (unit) circle
|-
| <source lang=apl inline>~</source>
| &quot;not&quot; and &quot;but not&quot;
|-
| <source lang=apl inline>≠</source>
| monadic means &quot;not equal to any preceding item&quot;
|-
| <source lang=apl inline>⍴</source>
| shape and reshape
|-
| <source lang=apl inline>⍳</source>
| indices of and index of
|-
| <source lang=apl inline>⌽</source>
| rotate and mirror along last axis
|-
| <source lang=apl inline>⊖</source>
| rotate and mirror along first axis
|-
| <source lang=apl inline>⍕</source>
| format and format with specification
|}


{{APL development}}{{APL glyphs}}
{{APL development}}{{APL glyphs}}
[[Category:APL character set]]
[[Category:APL character set]]
Retrieved from ‘https://aplwiki.com/wiki/Special:MobileDiff/8881’

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