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{{Built-in|Equal to|<nowiki>=</nowiki>}} is a [[comparison function]] which tests whether argument elements are [[Tolerant comparison|tolerantly]] equal to each other: it returns 1 if the [[element]]s being compared [[match]] and 0 if they do not. It is the [[Not|negation]] of [[Not Equal to]] (<syntaxhighlight lang=apl inline>≠</ | {{Built-in|Equal to|<nowiki>=</nowiki>}} is a [[comparison function]] which tests whether argument elements are [[Tolerant comparison|tolerantly]] equal to each other: it returns 1 if the [[element]]s being compared [[match]] and 0 if they do not. It is the [[Not|negation]] of [[Not Equal to]] (<syntaxhighlight lang=apl inline>≠</syntaxhighlight>). | ||
APL's <syntaxhighlight lang=apl inline>=</ | APL's <syntaxhighlight lang=apl inline>=</syntaxhighlight> differs from the usage of <math>=</math> in [[Comparison with traditional mathematics|traditional mathematical notation]] by having an implied [[Ken_Iverson#Iverson_bracket|Iverson bracket]]. Thus, <syntaxhighlight lang=apl inline>i=j</syntaxhighlight> is APL is equivalent to <math>[i=j]</math> or <math>\delta_{ij}</math> ([[wikipedia:Kronecker delta|Kronecker delta]]) in mathematics. | ||
== Examples == | == Examples == | ||
| Line 13: | Line 13: | ||
3j2 2j2 1j2 = 2j2 | 3j2 2j2 1j2 = 2j2 | ||
0 1 0 | 0 1 0 | ||
</ | </syntaxhighlight> | ||
It is subject to [[comparison tolerance]]: floating-point numbers which are very close together are considered equal, even though they are slightly different: | It is subject to [[comparison tolerance]]: floating-point numbers which are very close together are considered equal, even though they are slightly different: | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
| Line 25: | Line 25: | ||
1 {⎕CT←0⋄⍺=⍵} x ⍝ Overriding ⎕CT within a dfn | 1 {⎕CT←0⋄⍺=⍵} x ⍝ Overriding ⎕CT within a dfn | ||
0 0 1 0 0 | 0 0 1 0 0 | ||
</ | </syntaxhighlight> | ||
In a [[Nested array model|nested]] APL, Equal to is pervasive, and compares all [[simple scalar]]s found in a [[nested array]]. | In a [[Nested array model|nested]] APL, Equal to is pervasive, and compares all [[simple scalar]]s found in a [[nested array]]. | ||
<syntaxhighlight lang=apl> | <syntaxhighlight lang=apl> | ||
| Line 40: | Line 40: | ||
│└───┴─┘│ │ │ | │└───┴─┘│ │ │ | ||
└───────┴─────┴───┘ | └───────┴─────┴───┘ | ||
</ | </syntaxhighlight> | ||
In a [[Flat array model|flat]] array language such as [[J]], it instead compares [[Box]]es directly, since they are the [[element]]s of a boxed array. | In a [[Flat array model|flat]] array language such as [[J]], it instead compares [[Box]]es directly, since they are the [[element]]s of a boxed array. | ||
<syntaxhighlight lang=j> | <syntaxhighlight lang=j> | ||
(<'string') = 'which';'is';'string';'?' | (<'string') = 'which';'is';'string';'?' | ||
0 0 1 0 | 0 0 1 0 | ||
</ | </syntaxhighlight> | ||
== Boolean function == | == Boolean function == | ||
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When the arguments to Equal to are [[Boolean]], it is the [[wikipedia:Logical biconditional|Logical biconditional]] function, or the [[Not|negation]] of the [[xor]] function (which is [[Not Equal to]] restricted to Boolean arguments): | When the arguments to Equal to are [[Boolean]], it is the [[wikipedia:Logical biconditional|Logical biconditional]] function, or the [[Not|negation]] of the [[xor]] function (which is [[Not Equal to]] restricted to Boolean arguments): | ||
{|class=wikitable | {|class=wikitable | ||
!<syntaxhighlight lang=apl inline>=</ | !<syntaxhighlight lang=apl inline>=</syntaxhighlight>!!<syntaxhighlight lang=apl inline>0</syntaxhighlight>!!<syntaxhighlight lang=apl inline>1</syntaxhighlight> | ||
|- | |- | ||
!<syntaxhighlight lang=apl inline>0</ | !<syntaxhighlight lang=apl inline>0</syntaxhighlight> | ||
|<syntaxhighlight lang=apl inline>1</ | |<syntaxhighlight lang=apl inline>1</syntaxhighlight>||<syntaxhighlight lang=apl inline>0</syntaxhighlight> | ||
|- | |- | ||
!<syntaxhighlight lang=apl inline>1</ | !<syntaxhighlight lang=apl inline>1</syntaxhighlight> | ||
|<syntaxhighlight lang=apl inline>0</ | |<syntaxhighlight lang=apl inline>0</syntaxhighlight>||<syntaxhighlight lang=apl inline>1</syntaxhighlight> | ||
|} | |} | ||
In the context of logic, it can be read as ''if and only if''. This function is also known as xnor or the Kronecker delta. | In the context of logic, it can be read as ''if and only if''. This function is also known as xnor or the Kronecker delta. | ||
[[Reduction]] with <syntaxhighlight lang=apl inline>=</ | [[Reduction]] with <syntaxhighlight lang=apl inline>=</syntaxhighlight> is the same as reduction with xor (<syntaxhighlight lang=apl inline>≠</syntaxhighlight>), but [[Not|inverted]] if the number of reduced elements is even. The left and right [[identity element]] for Boolean <syntaxhighlight lang=apl inline>=</syntaxhighlight> is 1. | ||
== External links == | == External links == | ||