See the core module List::Util, especially all.

use strict; use warnings;
use List::Util 'all';

# see below for how to properly create your hash
my %words = (
    foo => 1,
    bar => 1,
    baz => 1,
    qux => 1,
);

my @must_exist = (
    'bar',
    'qux',
);

my $all_found = all { exists $words{ $_ } } @must_exist;

if ( $all_found ) {
    ...
}
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As for the code you have:

• my %hash = @allwords; #puts array in hash to search / compare.
  [download]

  ... yes, but using pairs. So an array of ('foo', 'bar', 'baz', 'qux') will give a hash with only two key-value pairs, with keys foo and baz. Thus later searching on the keys will not give the desired results.

  If you do want to create a hash with a key for each element in a list, use map:

  @array = ('foo', 'bar', 'baz', 'qux');
  %hash = map { $_ => 1 } @array;
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• if (exists $hash{'word1' && 'word2' && 'word3'}) {
     ...do stuff ;
  }
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  ... this won't work. You need to check whether each key exists individually:

  if ( exists $hash{'foo'} and exists $hash{'bar'} ) {
      ...
  }
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• if ( grep { $_ eq 'word1' && 'word2' } @allwords ) {
      ...do stuff; 
  }
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  ... this has several problems, which I will let other monks explain, but all of which can be avoided by using all.

Hope this helps!

update: added comments on OP code, then example with all