Working with sets

In all examples of sets given so far sets are used for indexing other modeling objects. But they may also be used for different purposes.

The following example (model setops.mos) demonstrates the use of basic set operations in Mosel: union (+), intersection (*), and difference (-):

model "Set example"

 declarations
  Cities={"rome", "bristol", "london", "paris", "liverpool"}
  Ports={"plymouth", "bristol", "glasgow", "london", "calais",
              "liverpool"}
  Capitals={"rome", "london", "paris", "madrid", "berlin"}
 end-declarations

 Places:= Cities+Ports+Capitals        ! Create the union of all 3 sets
 In_all_three:= Cities*Ports*Capitals  ! Create the intersection of all 3 sets
 Cities_not_cap:= Cities-Capitals      ! Create the set of all cities that are
                                       ! not capitals

 writeln("Union of all places: ", Places)
 writeln("Intersection of all three: ", In_all_three)
 writeln("Cities that are not capitals: ", Cities_not_cap)

end-model

The output of this example will look as follows:

  Union of all places:{`rome',`bristol',`london',`paris',`liverpool',
  `plymouth',`bristol',`glasgow',`calais',`liverpool',`rome',`paris',
  `madrid',`berlin'}
  Intersection of all three: {`london'}
  Cities that are not capitals: {`bristol',`liverpool}

Sets in Mosel are indeed a powerful facility for programming as in the following example (model prime.mos) that calculates all prime numbers between 2 and some given limit.

Starting with the smallest one, the algorithm takes every element of a set of numbers SNumbers (positive numbers between 2 and some upper limit that may be specified when running the model), adds it to the set of prime numbers SPrime and removes the number and all its multiples from the set SNumbers.

model Prime

 parameters
  LIMIT=100                     ! Search for prime numbers in 2..LIMIT
 end-parameters

 declarations
  SNumbers: set of integer      ! Set of numbers to be checked
  SPrime: set of integer        ! Set of prime numbers
 end-declarations

 SNumbers:={2..LIMIT}

 writeln("Prime numbers between 2 and ", LIMIT, ":")

 n:=2
 repeat
   while (not(n in SNumbers)) n+=1
   SPrime += {n}                ! n is a prime number
   i:=n
   while (i<=LIMIT) do          ! Remove n and all its multiples
     SNumbers-= {i}
     i+=n
   end-do
 until SNumbers={}

 writeln(SPrime)
 writeln(" (", getsize(SPrime), " prime numbers.)")

end-model

This example uses a new function, getsize, that if applied to a set returns the number of elements of the set. The condition in the while loop is the logical negation of an expression, marked with not: the loop is repeated as long as the condition n in SNumbers is not satisfied.

Set operators

The preceding example introduces the operator += to add sets to a set (there is also an operator -= to remove subsets from a set). Another set operator used in the example is in denoting that a single object is contained in a set. We have already encountered this operator in the enumeration of indices for the forall loop.

Mosel also defines the standard operators for comparing sets: subset (<=), superset (>=), difference (<>), end equality (=). Their use is illustrated by the following example (model setcomp.mos):

model "Set comparisons"

 declarations
  RAINBOW = {"red", "orange", "yellow", "green", "blue", "purple"}
  BRIGHT = {"yellow", "orange"}
  DARK = {"blue", "brown", "black"}
 end-declarations

 writeln("BRIGHT is included in RAINBOW: ", BRIGHT <= RAINBOW)
 writeln("RAINBOW is a superset of DARK: ", RAINBOW >= DARK)
 writeln("BRIGHT is different from DARK: ", BRIGHT <> DARK)
 writeln("BRIGHT is the same as RAINBOW: ", BRIGHT = RAINBOW)

end-model

As one might have expected, this example produces the following output:

BRIGHT is included in RAINBOW: true
RAINBOW is a superset of DARK: false
BRIGHT is different from DARK: true
BRIGHT is the same as RAINBOW: false