How to See the Equivalence of Select and Cases

MatchQ and PatternTest Equate Cases and Select

Select and Cases are essential Mathematica filtering functions. Select filters expressions, usually Lists, with predicates, such as ones that end with Q (evaluate ?*Q to see them) or those that don't, such as Greater, Less, and others that I list here. Cases uses patterns such as testing for a Head (e.g. _Integer) or a more general pattern (e.g. for a 2-element List, {x_,y_}.

Cases and Select are equivalent and a versatile Mathematica programmer knows how to use either one in any circumstance according to the need of the moment. The keys to seeing their equivalence are MatchQ, which translates a pattern into a predicate, and PatternTest (_?test), which translates a predicate into a pattern.

MatchQ[4.1,_Real]

True

MatchQ[{2,2.1},{_Integer,_Real}]

True

Thus you can take any usage of Cases and translate it into Select.

Cases[{a,4,4.2,6,7.5,8},x_Integer/;x>4]

{6,8}

integerGreaterThan4Q@n_:=MatchQ[n,_Integer&&n>4];

Select[{a,4,4.2,6,7.5,8},integerGreaterThan4Q]

{6,8}

And you can take any usage of Select and translate it into Cases.

Select[{1,2,4,7,6,2},EvenQ]

{2,4,6,2}

Cases[{1,2,4,7,6,2},_?EvenQ]

{2,4,6,2}

If we built our own predicate for EvenQ using MatchQ, it works the same way.

Select[{1,2,3,4},MatchQ[#/2,_Integer]&]

{2,4}

However note the need for parentheses around the test, which is often the case with PatternTest - the no parens usage fails.

Cases[{1,2,3,4},_?(MatchQ[#/2,_Integer]&)]

{2,4}

Cases[{1,2,3,4},_?MatchQ[#/2,_Integer]&]

{}

The reason is that Blank and PatternTest both have a higher Precedence ('stickiness') than Function, and so they stick together and PatternTest ends up inside the pure Function instead of using the pure Function as the pattern test. You don't need to worry about that — just as a rule of thumb put parens around any 'homemade' test.

Precedence/@{PatternTest,Blank,Function}

{680.,730.,90.}

The Q Predicates

This is a handy way to use Partition, especially using {} so a final short sublist won't be dropped from a 'ragged' List. The syntax says

Partition[list, sublist length, offset length, start Position, padding]

and here means divide the List into Partitions of length 4, moving the window that same length 4 with each Partition so there is no overlap, start the List at Position 1,  and the empty set {} for padding tells Partition to return the final sublist even if it's shorter than what is specified by the 2nd argument, and don't pad that final short list. Evaluate this code in your Notebook:

Qpredicates=Names@"*Q";

Print@Length@Qpredicates;

Partition[Qpredicates,4,4,1,{}]//TableForm

Here's the same result in a List.

Names@"*Q"

{AcyclicGraphQ,AlgebraicIntegerQ,AlgebraicUnitQ,AntihermitianMatrixQ,AntisymmetricMatrixQ,ArgumentCountQ,ArrayQ,AssociationQ,AtomQ,BinaryImageQ,BipartiteGraphQ,BooleanQ,BoundaryMeshRegionQ,BoundedRegionQ,BusinessDayQ,ColorQ,CompatibleUnitQ,CompleteGraphQ,CompositeQ,ConnectedGraphQ,ConstantRegionQ,ContinuousTimeModelQ,ControllableModelQ,CoprimeQ,DateObjectQ,DaylightQ,DayMatchQ,DeviceOpenQ,DiagonalizableMatrixQ,DigitQ,DirectedGraphQ,DirectoryQ,DiscreteTimeModelQ,DisjointQ,DispatchQ,DistributionParameterQ,DuplicateFreeQ,EdgeCoverQ,EdgeQ,EllipticNomeQ,EmptyGraphQ,EulerianGraphQ,EvenQ,ExactNumberQ,FileExistsQ,firedQ,FreeQ,GeoWithinQ,GraphQ,GroupElementQ,HamiltonianGraphQ,HermitianMatrixQ,HypergeometricPFQ,ImageQ,IndefiniteMatrixQ,IndependentEdgeSetQ,IndependentVertexSetQ,InexactNumberQ,integerGreaterThan4Q,IntegerQ,IntersectingQ,IntervalMemberQ,InverseEllipticNomeQ,IrreduciblePolynomialQ,IsomorphicGraphQ,KEdgeConnectedGraphQ,KeyExistsQ,KeyFreeQ,KeyMemberQ,KnownUnitQ,KVertexConnectedGraphQ,LeapYearQ,LegendreQ,LetterQ,LinkConnectedQ,LinkReadyQ,ListQ,LoopFreeGraphQ,LowerCaseQ,MachineNumberQ,ManagedLibraryExpressionQ,MandelbrotSetMemberQ,MarcumQ,MatchLocalNameQ,MatchQ,MatrixQ,MemberQ,MeshRegionQ,MixedGraphQ,MultigraphQ,NameQ,NegativeDefiniteMatrixQ,NegativeSemidefiniteMatrixQ,NormalMatrixQ,NumberQ,NumericQ,ObservableModelQ,OddQ,OptionQ,OrderedQ,OrthogonalMatrixQ,OutputControllableModelQ,PartitionsQ,PathGraphQ,PermutationCyclesQ,PermutationListQ,PlanarGraphQ,PolynomialQ,PositiveDefiniteMatrixQ,PositiveSemidefiniteMatrixQ,PossibleZeroQ,PrimePowerQ,PrimeQ,ProcessParameterQ,QHypergeometricPFQ,QuadraticIrrationalQ,QuantityQ,RegionQ,RegularlySampledQ,RootOfUnityQ,SameQ,SatisfiableQ,ScheduledTaskActiveQ,SimpleGraphQ,SquareFreeQ,SquareMatrixQ,StringFreeQ,StringMatchQ,StringQ,SubsetQ,SymmetricMatrixQ,SyntaxQ,TautologyQ,TensorQ,TreeGraphQ,TrueQ,UnateQ,UndirectedGraphQ,UnitaryMatrixQ,UnsameQ,UpperCaseQ,URLExistsQ,ValueQ,VectorQ,VertexCoverQ,VertexQ,WeaklyConnectedGraphQ,WeightedGraphQ}

 

Getting Help: Find a Function Using Wildcards

There are over 4000 built-in functions in Mathematica, not including those in add-on Packages. 

In[340]:= Names["System`*"]//Length

Out[340]= 4131

Therefore it makes sense to see if there is a built-in function before we write one. Likewise, the large number of functions means that we often have a vague memory of a function we've used but can't put our finger on its name when we need it again. If I have some idea of the name of a function I've used, or want to see if there is an existing function and guess at part of its name, I use a wildcard in my current Notebook with the short form of Information (?). To find a function or survey related functions, this is faster than using the Doc Center.

Here I want to know the filepath of my current Notebook, which I cannot find in any Mathematica menu item. Maybe there's a Notebook function to tell me. I scan down the results until I see NotebookFileName, which is the one I want. I also quickly click on a half-dozen or so functions to refresh my memory of them or explore new ones and read the short info displayed below the results.

Using Information this way is always fruitful for me since I efficiently learn about functions I didn't remember or didn't know. But if it fails to flush out the function I want, I go to the Doc Center and search there.