Adding to immutable HashSet

You are not adding to the HashSet. You are assigning to hashSet, which is perfectly fine, since hashSet is a var, not a val.

Section 6.12.4 Assignment Operators of the Scala Language Specification (SLS) explains how such compound assignment operators are desugared:

l ω= r

(where ω is any sequence of operator characters other than <, >, ! and doesn't start with =) gets desugared to

l.ω=(r)

iff l has or is implicitly convertible to an object that has a member named ω=.

Otherwise, it gets desugared to

l = l.ω(r)

(except l is guaranteed to be only evaluated once), if that typechecks.

This allows something like += to work like it does in other languages but still be overridden to do something different.

Observe this:

scala> var hashSet: Set[Int] = new collection.immutable.HashSet
hashSet: Set[Int] = Set()

scala> val set2 = hashSet + 1234
set2: scala.collection.immutable.Set[Int] = Set(1234)

scala> set2
res20: scala.collection.immutable.Set[Int] = Set(1234)

scala> hashSet
res21: Set[Int] = Set()

So nothing gets added to the immutable hashSet. hashSet is the same as it was when constructed. + returns a new set altogether and the original set is unchanged.

When you do hashSet += 1234, it is a scala shorthand for (note no method += exists in HashSet):

val temp = hashSet + 1234
hashSet = temp

+= will work for any class which follows this protocol. In short when you do a += 12. a must have a method + which returns the same type as a's and a should be assignable (i.e. a var. It does not work for val. Try this: val i = 23; i+=1).

Short answer

You have a var so you can reassign to it. So += in this case will be translated to

hashSet = hashSet + elem

just like other types, as long as + is defined on them

var i = 0
i += 1
i = i + 1

Details

immutable.HashSeth has + method which

Creates a new set with an additional element, unless the element is already present.

according to docs.

There is no += method defined in this class, so += will be a synthetic method given to you by compiler which acts as an operator simply calling the + method on the left operand by passing the right operand and assigning the result back to the left operand.