How does HashSet not allow duplicates?

See HashMap#put:

Associates the specified value with the specified key in this map. If the map previously contained a mapping for the key, the old value is replaced.

It replaces the key with the new value, this way, no duplicates will be in the HashSet.

PRESENT is just a dummy value -- the set doesn't really care what it is. What the set does care about is the map's keys. So the logic goes like this:

Set.add(a):
  map.put(a, PRESENT) // so far, this is just what you said
    the key "a" is in the map, so...
      keep the "a" key, but map its value to the PRESENT we just passed in
      also, return the old value (which we'll call OLD)
  look at the return value: it's OLD, != null. So return false.

Now, the fact that OLD == PRESENT doesn't matter -- and note that Map.put doesn't change the key, just the value mapped to that key. Since the map's keys are what the Set really cares about, the Set is unchanged.

In fact, there has been some change to the underlying structures of the Set -- it replaced a mapping of (a, OLD) with (a, PRESENT). But that's not observable from outside the Set's implementation. (And as it happens, that change isn't even a real change, since OLD == PRESENT).

As you can see the HashSet.add method adds the element to the HashMap.put as a key not as a value. Value is replaced in the HashMap not the key.

The answer that you may be looking comes down to the fact that the backing hashmap maps the elements of the set to the value PRESENT which is defined in HashSet.java as follows:

private static final Object PRESENT = new Object();

In the source code for HashMap.put we have:

  386     public V put(K key, V value) {
  387         if (key == null)
  388             return putForNullKey(value);
  389         int hash = hash(key.hashCode());
  390         int i = indexFor(hash, table.length);
  391         for (Entry<K,V> e = table[i]; e != null; e = e.next) {
  392             Object k;
  393             if (e.hash == hash && ((k = e.key) == key || key.equals(k))) {
  394                 V oldValue = e.value;
  395                 e.value = value;
  396                 e.recordAccess(this);
  397                 return oldValue;
  398             }
  399         }
  400 
  401         modCount++;
  402         addEntry(hash, key, value, i);
  403         return null;
  404     }

Because the key in question already exists, we will take the early return on line 397. But you might think a change is being made to the map on line 395, in which it appears that we are changing the value of a map entry. However, the value of value is PRESENT. But because PRESENT is static and final, so there is only one such instance; and so the assignment e.value = value actually doesn't change the map, and therefore the set, at all!

Update:

Once a HashSet is initialized.
- All the items in it are stored as keys in a HashMap
- All the values that HashMap have ONLY ONE object that is PRESENT which is a static field in HashSet