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:
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}
