Adds set algebra laws

Author: JJ

doc/Language/math.pod6

@@ -1,26 +1,62 @@
 =begin pod :tag<perl6>
 
-=TITLE Interacting with the system
+=TITLE Doing math with Perl 6
 
-=SUBTITLE How to work with other applications and the operating system at large
+=SUBTITLE Different mathematical paradigms and how they are implemented in this language.
 
-=head1 Running external programs.
+=head1 Sets.
 
-TBD
+Perl 6 includes the L<Set> data type, as well as support for L<most set operations|/language/setbagmix#Set/Bag_Operators>. L<Union and intersection|https://en.wikipedia.org/wiki/Algebra_of_sets> are not only native operations, they use their I<natural> symbols, ∩ and ∪. For instace, this code would check the fundamental laws of the arithmetic of sets for a limited number of sets:
 
-=head1 Interacting with the system in an asynchronous way
+=begin code
+my @arbitrary-numbers = ^100;
+my \U = @arbitrary-numbers.Set;
+my \Ø = ().Set;
 
-TBD
+my @sets;
+
+@sets.push: Set.new( @arbitrary-numbers.pick( @arbitrary-numbers.elems.rand)) for @arbitrary-numbers;
+
+my (@union, @intersection);
+
+for @sets -> $set {
+    @union.push: $set ∩ $set === $set;
+    @intersection.push: $set ∪ $set === $set;
+}
+
+say "Idempotent union is ", so @union.all;
+# OUTPUT: «Idempotent union is True»
+say "Idempotent intersection is ", so @intersection.all;
+# OUTPUT: «Idempotent intersection is True»
+my (@universe, @empty-set, @id-universe, @id-empty);
 
-=head1 Using the native interface
+for @sets -> \A {
+    @universe.push: A ∪ U === U;
+    @id-universe.push: A ∩ U === A;
+    @empty-set.push: A ∩ Ø === Ø;
+    @id-empty.push: A ∪ Ø === A;
+}
+
+say "Universe dominates ", so @universe.all;    # OUTPUT: «Universe dominates True»
+say "Empty set dominates ", so @empty-set.all;  # OUTPUT: «Empty set dominates True»
+
+say "Identity with U ", so @id-universe.all;    # OUTPUT: «Identity with U True»
+say "Identity with Ø ", so @id-empty.all;       # OUTPUT: «Identity with Ø True»
+=end code
+
+In this code, not only we check if the equalities in the algebra of sets hold, we also use, via L<sigilless variables|/language/variables#index-entry-\_(sigilless_variables)> and the Unicode form of the set operators, expressions that are as close as possible to the original form; C<A ∪ U === U>, for example, except for the use of the L<value identity operator <===>|/routine/===> is very close to the actual mathematical expression in the L<Wikipedia entry|https://en.wikipedia.org/wiki/Algebra_of_sets>.
+
+=head1 Arithmetic.
 
 TBD
 
-=head1 Defining your own associative structures
+=head1 Sequences
 
 TBD
 
+=head1 Mathematical constants
 
+TBD
 
 
 =end pod