I don't know what underlying perl mechanics I'm potentially violating (update: I meant in generating union, everything else should be jake), but the following seems to work. There's no idea here that wasn't in your original post; in fact I referred to it frequently when writing this up.

#!/usr/bin/env perl

use strict;
use warnings;

my @array1=(1..5);
my @array2=(3..7);

my (%union, %sdiff, %sd1, %sd2, %inter);

@union{(@array1,@array2)} = (@array1,@array2);

%sd1 = %sd2 = %inter = %union;

delete @sd1{@array2};
delete @sd2{@array1};
%sdiff = (%sd1,%sd2);

delete @inter{keys %sdiff};

print "array1: ", join (" ", @array1), "\n";
print "array2: ", join (" ", @array2), "\n";
print "union: ", join (" ", keys %union), "\n";
print "sdiff: ", join (" ", keys %sdiff), "\n";
print "inter: ", join (" ", keys %inter), "\n";
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It felt like I was writing either haiku or APL, but I can't quite say which....