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The [[bound]] of an array is the number of [[element]]s it contains, or equivalently the length ([[Tally]]) of its [[ravel]] vector or the product (< | The [[bound]] of an array is the number of [[element]]s it contains, or equivalently the length ([[Tally]]) of its [[ravel]] vector or the [[product]] of its [[shape]] (<syntaxhighlight lang=apl inline>×/⍴</syntaxhighlight>). | ||
The bound can be derived axiomatically as a function of the [[shape]] from two rules: | The bound can be derived axiomatically as a function of the [[shape]] from two rules: | ||
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* If two shapes are [[catenate]]d (for instance by an [[outer product]]), the resulting bound is the product of their individual bounds. | * If two shapes are [[catenate]]d (for instance by an [[outer product]]), the resulting bound is the product of their individual bounds. | ||
A scalar has [[rank]] 0, or empty shape. From the above axioms we can deduce that it has bound 1: catenating < | A scalar has [[rank]] 0, or empty shape. From the above axioms we can deduce that it has bound 1: catenating <syntaxhighlight lang=apl inline>⍬</syntaxhighlight> with some other shape leaves that shape unchanged, so multiplying any bound by a scalar's bound cannot change it either. Therefore a scalar's bound must be the multiplicative identity, 1. | ||
Combining [[Axis|axes]] of an array, for instance by using [[Table]] or [[Ravel]] with [[Function axis|axis]], leaves the array's bound unchanged, despite changing its shape, rank, and possibly [[Tally]]. | Combining [[Axis|axes]] of an array, for instance by using [[Table]] or [[Ravel]] with [[Function axis|axis]], leaves the array's bound unchanged, despite changing its shape, rank, and possibly [[Tally]]. | ||
{{APL | {{APL features}}[[Category:Array characteristics]] | ||