Merging two Maps

static void mergeSet(Map<String, Set<String>> map1, Map<String, Set<String>> map2) {
    map1.forEach((key1, value1) -> {
        map2.merge(key1, value1, (key2, value2) -> key2).addAll(value1);
    });
}

You can do this with a stream fairly easily:

Map<T, Set<U>> merged = Stream.of(first, second)
        .map(Map::entrySet)
        .flatMap(Set::stream)
        .collect(Collectors.toMap(Entry::getKey, Entry::getValue, (a, b) -> {
            HashSet<U> both = new HashSet<>(a);
            both.addAll(b);
            return both;
        }));

This splits the maps into their Entrys and then joins them with a Collector which resolves duplicates by adding both values to a new HashSet.

This also works for any number of maps.

Some variations which produce the same result:

Stream.of(first, second).flatMap(m -> m.entrySet().stream())
    .collect(...);
Stream.concat(first.entrySet().stream(), second.entrySet().stream())
    .collect(...); //from comment by Aleksandr Dubinsky

The third parameter for Collectors.toMap is not necessary if there are no duplicate keys.

There is another Collectors.toMap with a fourth parameter that lets you decide the type of the Map collected into.

Are we talking about HashMap instances. In that case lookup is O(1), so you can just take one map, iterate over the entries of that map, see whether the other map contains that key. If not, just add the set. If it contains the key, take the union of the two sets (by adding all elements of one set to another)

To illustrate with some code, where I used a Set to have autocompletion in my IDE

Map<String, Set<Double>> firstMap = new HashMap<String, Set<Double>>(  );
Map<String, Set<Double>> secondMap = new HashMap<String, Set<Double>>(  );
Set<Map.Entry<String, Set<Double>>> entries = firstMap.entrySet();
for ( Map.Entry<String, Set<Double>> entry : entries ) {
  Set<Double> secondMapValue = secondMap.get( entry.getKey() );
  if ( secondMapValue == null ) {
    secondMap.put( entry.getKey(), entry.getValue() );
  }
  else {
    secondMapValue.addAll( entry.getValue() );
  }
}