Solver Set Union, Intersection, and Relative Complement

Source code of 'Set Union, Intersection, and Relative Complement'

This Solver (Set Union, Intersection, and Relative Complement) was created by by jim_thompson5910(35256)

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==section input
This solver will show you how to perform a <a href=http://en.wikipedia.org/wiki/Set_(mathematics)>set</a> <a href=http://en.wikipedia.org/wiki/Union_(set_theory)>union</a>, a set <a href=http://en.wikipedia.org/wiki/Intersection_(set_theory)>intersection</a>, or a <a href=http://en.wikipedia.org/wiki/Complement_(set_theory)#Relative_complement>relative complement</a> on two sets.

Enter two sets. You can do so by entering each element of each set separated by commas:

A = {*[input 30 set1=a,b,c,d]} 
B = {*[input 30 set2=d,e,f]}

What do you want to do to these two sets (choose from set union, set intersection, or relative complement)?

*[choice operation union intersection complement]

Note: With the <a href=http://en.wikipedia.org/wiki/Complement_(set_theory)#Relative_complement>relative complement</a> option, it will give you all the elements in set B but NOT in set A (ie think of 'subtracting' out the elements of A from B). If you think of set B as the set U (the universal set), then you'll find the <a href=http://en.wikipedia.org/wiki/Complement_(set_theory)>complement</a> of set A.

==section solution

perl

#### Subroutines

sub remove_duplicates
{
 my @set=@_; 
 my @ans=(); 
 for(my $i=0;$i<@set;$i++)
 {
 for(my $j=0;$j<@set;$j++)
 {
 if($i!=$j)
 {
 $set[$j]="?" if $set[$i] eq $set[$j];
 }
 }
 }

for(my $i=0;$i<@set;$i++)
 {
 push(@ans,$set[$i]) if($set[$i]!~m/\?/); 
 } 
 return @ans;
}

sub union
{
 my ($set1,$set2)=@_;
 my @set1=split(/,/,$set1);
 my @set2=split(/,/,$set2);
 my @ans=@set1;
 
 for(my $i=0;$i<@set2;$i++)
 {
 push(@ans,$set2[$i]);
 }
 return @ans;
}

sub intersection
{
 my ($set1,$set2)=@_;
 my @set1=split(/,/,$set1);
 my @set2=split(/,/,$set2);
 my @ans=();
 
 for(my $i=0;$i<@set1;$i++)
 {
 for(my $j=0;$j<@set2;$j++)
 {
 if($set1[$i] eq $set2[$j])
 {
 push(@ans,$set1[$i]);
 }
 }
 }
 return @ans;
}

### given two sets A, B returns B \ A (ie all elements that are in B but NOT in A)
sub complement
{
    my ($set1,$set2)=@_;
    my @set1=split(/,/,$set1);
    my @set2=split(/,/,$set2);
    my @ans=();

## find intersection of A and B
    my @intersection=intersection($set1,$set2);

for(my $i=0;$i<@set2;$i++)
 {
 for(my $j=0;$j<@intersection;$j++)
 {
 if($set2[$i] eq $intersection[$j])
 {
 $set2[$i]="?";
 }
 }
 }

for(my $i=0;$i<@set2;$i++)
 {
 push(@ans,$set2[$i]) if $set2[$i]!~m/\?/;
 }
 return @ans;
}

### End Subroutines

$set1=~s/\s//g;
$set1=~s/^,//;
$set1=~s/,$//;
$set2=~s/\s//g;
$set2=~s/^,//;
$set2=~s/,$//;

### Error Checking
my $error_check1=$set1;
$error_check1=~s/\-|\/|\.|,//g;
if($error_check1=~m/\W/)
{
 print "Error: Invalid Symbol";
 goto end;
}

my $error_check2=$set2;
$error_check2=~s/\-|\/|\.|,//g;
if($error_check2=~m/\W/)
{
 print "Error: Invalid Symbol";
 goto end;
}

if($error_check1 eq '' or $error_check2 eq '')
{
 print "Error: One of the sets is empty. Please enter values.";
 goto end;
}

### End Error Checking

### Check for any duplicates before we get started
my $repeat_flag=0;
my @starter_set1=split(/,/,$set1);
my @ender_set1=remove_duplicates(@starter_set1);
if(@ender_set1 < @starter_set1)
{
 $repeat_flag=1;
 print " Note: Since a <a href=http://en.wikipedia.org/wiki/Set_(mathematics)>set</a> is a collection of distinct (ie different) objects, this means that duplicate elements are unnecessary. So let's remove the duplicates from the set {$set1} to get {".join(",",@ender_set1)."} "; 
 $set1=join(",",@ender_set1);
}

my @starter_set2=split(/,/,$set2);
my @ender_set2=remove_duplicates(@starter_set2);
if(@ender_set2 < @starter_set2)
{
 if($repeat_flag==0)
 {
 print " Note: Since a <a href=http://en.wikipedia.org/wiki/Set_(mathematics)>set</a> is a collection of distinct (ie different) objects, this means that duplicate elements are unnecessary. So let's remove the duplicates from the set {$set2} to get {".join(",",@ender_set2)."} "; 
 }
 else
 {
 print " Let's also remove the duplicates from {$set2} to get {".join(",",@ender_set2)."} "; 
 }
 $set2=join(",",@ender_set2);
}

print " Let's first write down the given sets (and set them equal to A and B):*[Tex \\LARGE A=\\left\\{$set1\\right\\}] and *[Tex \\LARGE B=\\left\\{$set2\\right\\}] ";

#### Set Union
if($operation eq "union")
{
my @temp=union($set1,$set2);
my @union=remove_duplicates(@temp);

print " The union of sets A and B will be denoted *[Tex \\LARGE A \\cup B] ";

print " To form the <a href=http://en.wikipedia.org/wiki/Union_(set_theory)>union</a> of the two sets A={$set1} and B={$set2}, simply make a new set that contains EVERY element in both sets like so {$set1,$set2}. Take note how the red elements are from the first set while the green elements are from the second set. ";

if(@temp > @union)
{
 print " Now remove any duplicate elements to get the set {".join(",",@union)."} ";
}

print " So if *[Tex \\LARGE A=\\left\\{$set1\\right\\}] and *[Tex \\LARGE B=\\left\\{$set2\\right\\}], then the union of the two sets is: *[Tex \\LARGE A \\cup B=\\left\\{".join(",",@union)."\\right\\}] ";

}

#### Set Intersection
if($operation eq "intersection")
{
my ($temp_set1,$temp_set2)=($set1,$set2);
my @temp_set1=split(/,/,$temp_set1);
my @temp_set2=split(/,/,$temp_set2);
my @temp=intersection($set1,$set2);
my @intersection=remove_duplicates(@temp);

for(my $i=0;$i<@temp_set1;$i++)
{
 for(my $j=0;$j<@intersection;$j++)
 {
 if($temp_set1[$i] eq $intersection[$j])
 {
 $temp_set1[$i]="$temp_set1[$i]";
 }
 } 
}
$temp_set1=join(",",@temp_set1);

for(my $i=0;$i<@temp_set2;$i++)
{
 for(my $j=0;$j<@intersection;$j++)
 {
 if($temp_set2[$i] eq $intersection[$j])
 {
 $temp_set2[$i]="$temp_set2[$i]"; 
 }
 } 
}
$temp_set2=join(",",@temp_set2);

print " The intersection of sets A and B will be denoted *[Tex \\LARGE A \\cap B] ";

print " To form the <a href=http://en.wikipedia.org/wiki/Intersection_(set_theory)>intersection</a> of the two sets A={$set1} and B={$set2}, simply write down all of the elements that are in BOTH sets at the same time. ";

print " So let's place the first set right above the second set (for an easy visual comparison): ";

print "{$set1} {$set2} ";

print " Now highlight the elements that are in BOTH sets: ";

print "{$temp_set1} {$temp_set2} ";

if(@intersection == 0)
{
 print " Since there are no common elements, this means that these two sets have nothing in common. So their intersection is the empty set (ie a set with no elements). Note: the empty set is denoted either by { } (set braces with nothing inside) or by *[Tex \\LARGE \\emptyset] (the empty set symbol). Furthermore, it is invalid to say *[Tex \\LARGE \\left\\{\\emptyset\\right\\}] ";
 #print " So if *[Tex \\LARGE A=\\left\\{$set1\\right\\}] and *[Tex \\LARGE B=\\left\\{$set2\\right\\}], then the intersection of the two sets is: *[Tex \\LARGE A \\cap B=\\\{ \\ \\\\}] ";

print " So if *[Tex \\LARGE A=\\left\\{$set1\\right\\}] and *[Tex \\LARGE B=\\left\\{$set2\\right\\}], then the intersection of the two sets is: *[Tex \\LARGE A \\cap B=\\emptyset] ";
 print " Since the intersection of these two sets is the empty set (ie the two sets have nothing in common), this means that the two sets are <a href=http://en.wikipedia.org/wiki/Disjoint_sets>mutually disjoint</a>. ";
 goto end;
}

print " Since the highlighted terms are ".join(",",@intersection).", this means that the intersection of the two sets is {".join(",",@intersection)."} ";

}

#### Set Complement
if($operation eq "complement")
{
 my @complement=complement($set1,$set2);
 my @intersection=intersection($set1,$set2);
 #print " First, write out all the elements in the second set B={$set2} ";
 
 my @temp_set2=split(/,/,$set2);
 for(my $i=0;$i<@temp_set2;$i++)
 {
 for(my $j=0;$j<@intersection;$j++)
 {
 if($temp_set2[$i] eq $intersection[$j])
 {
 $temp_set2[$i]="$temp_set2[$i]"; 
 }
 } 
 }
 
 #print " Now looking through the given set we've just written down, highlight all of the elements in set B that are also in set A={$set1}: {".join(",",@temp_set2)."} ";

print " The relative complement of A in B will be denoted *[Tex \\LARGE B \\setminus A] ";

print " Also, notice that the <a href=http://en.wikipedia.org/wiki/Intersection_(set_theory)>intersection</a> of the sets A and B is *[Tex \\LARGE A \\cap B =";

print "\\left\\{".join(",",@intersection)."\\right\\}" if(@intersection > 0);
print "\\emptyset" if(@intersection == 0);

print "]. You can use this solver to find the intersection of the sets A and B (just change the options above). ";

print " To find the relative complement of A in B, first write down the second set B: ";

print "B={$set2} ";

print " Now highlight the elements that are in the intersection (see above): ";

print "{".join(",",@temp_set2)."} ";

if(@intersection == 0)
 {
 print " Since there are no common elements between the two given sets, this means that *[Tex \\LARGE B \\setminus A=\\left\\{".join(",",@complement)."\\right\\}]. This also means that *[Tex \\LARGE B \\setminus A = B] ";
 goto end;
 }

print " Now simply remove the highlighted elements to get: {".join(",",@complement)."} ";

print " So if *[Tex \\LARGE A=\\left\\{$set1\\right\\}] and *[Tex \\LARGE B=\\left\\{$set2\\right\\}], then *[Tex \\LARGE B \\setminus A=\\left\\{".join(",",@complement)."\\right\\}] ";

}

end:

==section output

==section check