More Complex Predicates: allOddQ, allIdenticalQ, subsetQ

"Mr Wizard Todd", Manfred Plagmann, and Sophia Scheibe have done Mathematica users a nice favor by getting permission from McGraw-Hill to distribute licensed copies of David Wagner' s superb book, Power Programming in Mathematica. (Dropbox link: https://www.dropbox.com/s/j2dsyvptnxjd369/Wagner%20All%20Parts-RC.pdf)

A related post is How to See the Equivalence of Select and Cases.

Here are some nice examples of predicates from Wagner (re - written in my Prefix/Postfix dialect).Test a list to see if all of its entries are odd integers.

allOdd@aList_List:=Length@Select[aList,OddQ]==Length@aList

allOdd@{2,3,5,6}

False

allOdd@{3,5,7,9}

True

This predicate tests a List to see if its parts are identical. Count returns all Parts that are Equal to the First Part.

identicalListPartsQ@aList_List:=Count[aList, First@aList] == Length@aList

identicalListPartsQ@{a,b,c,4}

False

identicalListPartsQ@{a,a,a,a}

True

SameQ vs. Equal  and Testing for Lists

SubsetQ determines if its first argument is a subset of its second argument and returns True or False. Wagner's function did not include a List test and he notes on Equal vs.SameQ : "The use of === rather than == makes SubsetQ return False if either set1 or set2 is not a list. Try it with Equal."

I use the Head test: set1_List, which rejects a non-List argument before evaluation. And more importantly, SameQ should always be used to test non-numerical equivalence. In fact using Equal here can give screwy results in part because of its usage as the mathematical "equals" (=) in Mathematica's syntax for equations.

When using abbreviated operators, we should first determine their Precedence:

Precedence/@{Union,Equal,SameQ}

{300.,290.,290.}

Since Union is slightly stickier than SameQ, the Union will be performed before the SameQ test.

subsetQ[set1_,set2_]:=set1∪set2===set2

subsetQ[{a,b,c},{a,b,d}]

False

subsetQ[{a,b},{a,b,2}]

False

Union Sorts Its Result

Whoops! Wagner's function didn't work. Let's find out why:

subsetQ[{a,b},{a,b,2}]//Trace

{subsetQ[{a,b},{a,b,2}],{a,b}∪{a,b,2}==={a,b,2},{{a,b}∪{a,b,2},{2,a,b}},{2,a,b}==={a,b,2},False}

The fact that Union returns a sorted List fouls up the comparison. Let's try a fix with Sort:

Clear@subsetQ;subsetQ[set1_,set2_]:=set1∪set2===Sort@set2

And it works - you see in the last step Sort fixes the order.

subsetQ[{a,b},{a,b,2}]//Trace

{subsetQ[{a,b},{a,b,2}],{a,b}∪{a,b,2}===Sort[{a,b,2}],{{a,b}∪{a,b,2},{2,a,b}},{Sort[{a,b,2}],{2,a,b}},{2,a,b}==={2,a,b},True}

Greater (>) is a "non-Q" predicate so the last statement (lacking a return-suppressing semi-colon), returns True or False.

More on predicates: http://mathematica-guide.blogspot.com/2015/07/how-to-see-equivalence-of-select-and.html

Assignment to a Pattern

I'll use a couple of examples on searching Strings from Sal Mangano's excellent Mathematica Cookbook to show how to use Pattern Assignment. First, it's simple enough to search some Strings for those that are numbers:

StringMatchQ[{"12345", "2468", "abcde"}, NumberString]

{True, True, False}

But if you want to add a Predicate, you need another device such as naming a Pattern and using the name in the Predicate. This is an example of Pattern Assignment where the Pattern is named "m".

StringMatchQ[{"12345", "2468", "abcde"}, m : NumberString /; OddQ@FromDigits@m]

{True, False, False}

As Sal Mangone notes in his Mathematica Cookbook and I explore further in Predicates, Tests, and Test Patterns, adding a Predicate to a Pattern is a powerful device because we can include in the Predicate anything that Mathematica can compute. A Predicate can be a built-in one or one we construct, so the possibilities are truly limitless.