And: Difference between revisions

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(Move LCM to its own page; see Talk:Or)
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{{Built-in|And|∧}} is a [[dyadic]] [[scalar function|scalar]] [[boolean function]] which tests if both arguments are true: it returns 1 if both are 1 and 0 if one or both are 0. It represents the [[wikipedia:logical conjunction|logical conjunction]] in Boolean logic.
{{Built-in|And|∧}} is a [[dyadic]] [[scalar function|scalar]] [[boolean function]] which tests if both arguments are true: it returns 1 if both are 1 and 0 if one or both are 0. It represents the [[wikipedia:logical conjunction|logical conjunction]] in Boolean logic. In many APLs, And is a special case of the [[LCM]] function.


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0
0
</source>
</source>
== Extended definition ==
Many APL implementations extend this function to non-Boolean arguments. In this case, this function behaves as '''[[wikipedia:Least common multiple|Least Common Multiple]]''' or '''LCM'''. For positive integer arguments, it is defined as the smallest positive number which is divisible by both numbers. If one of the arguments is zero, the LCM function returns zero.
<source lang=apl>
      ∘.∧⍨ 0,⍳10
0  0  0  0  0  0  0  0  0  0  0
0  1  2  3  4  5  6  7  8  9 10
0  2  2  6  4 10  6 14  8 18 10
0  3  6  3 12 15  6 21 24  9 30
0  4  4 12  4 20 12 28  8 36 20
0  5 10 15 20  5 30 35 40 45 10
0  6  6  6 12 30  6 42 24 18 30
0  7 14 21 28 35 42  7 56 63 70
0  8  8 24  8 40 24 56  8 72 40
0  9 18  9 36 45 18 63 72  9 90
0 10 10 30 20 10 30 70 40 90 10
</source>{{Works in|[[Dyalog APL]]}}
While the mathematical definition of LCM does not cover non-integers, some implementations accept them as arguments. In this case, the return value of <source lang=apl inline>R←X∧Y</source> is chosen so that both <source lang=apl inline>R÷X</source> and <source lang=apl inline>R÷Y</source> are integers (or [[wikipedia:Gaussian integer|Gaussian integers]], when X and/or Y are [[complex]] numbers).
<source lang=apl>
      0.9∧25÷6
112.5
      112.5÷0.9(25÷6)
125 27
      2J2∧3J1
6J2
      6J2÷2J2 3J1
2J¯1 2
</source>{{Works in|[[Dyalog APL]]}}


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

Revision as of 15:09, 1 June 2020

And () is a dyadic scalar boolean function which tests if both arguments are true: it returns 1 if both are 1 and 0 if one or both are 0. It represents the logical conjunction in Boolean logic. In many APLs, And is a special case of the LCM function.

0 1
0 0 0
1 0 1

Examples

The following shows all possible combinations of inputs as a Boolean function.

      0 0 1 1 ∧ 0 1 0 1
0 0 0 1

When combined with Reduce, And can be used to test if every value in a Boolean vector is true.

      ∧/ 1 1 1 1 1
1
      ∧/ 1 0 0 1 1
0

External links

Documentation

APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic ConjugateNegateSignumReciprocalMagnitudeExponentialNatural LogarithmFloorCeilingFactorialNotPi TimesRollTypeImaginarySquare Root
Dyadic AddSubtractTimesDivideResiduePowerLogarithmMinimumMaximumBinomialComparison functionsBoolean functions (And, Or, Nand, Nor) ∙ GCDLCMCircularComplexRoot
Non-Scalar
Structural ShapeReshapeTallyDepthRavelEnlistTableCatenateReverseRotateTransposeRazeMixSplitEncloseNestCut (K)PairLinkPartitioned EnclosePartition
Selection FirstPickTakeDropUniqueIdentityStopSelectReplicateExpandSet functions (IntersectionUnionWithout) ∙ Bracket indexingIndexCartesian ProductSort
Selector Index generatorGradeIndex OfInterval IndexIndicesDealPrefix and suffix vectors
Computational MatchNot MatchMembershipFindNub SieveEncodeDecodeMatrix InverseMatrix DivideFormatExecuteMaterialiseRange
Operators Monadic EachCommuteConstantReplicateExpandReduceWindowed ReduceScanOuter ProductKeyI-BeamSpawnFunction axis
Dyadic BindCompositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner ProductDeterminantPowerAtUnderRankDepthVariantStencilCutDirect definition (operator)
Quad names Index originComparison toleranceMigration levelAtomic vector