It's written in all decent java courses, that if you implement the Comparable interface, you should (in most cases) also override the equals method to match its behavior.
Unfortunately, in my current organization people try to convince me to do exactly the opposite. I am looking for the most convincing code example to show them all the evil that will happen.
I think you can beat them by showing the Comparable javadoc that says:
It is strongly recommended (though not required) that natural orderings be consistent with equals. This is so because sorted sets (and sorted maps) without explicit comparators behave "strangely" when they are used with elements (or keys) whose natural ordering is inconsistent with equals. In particular, such a sorted set (or sorted map) violates the general contract for set (or map), which is defined in terms of the equals method.
For example, if one adds two keys a and b such that (!a.equals(b) && a.compareTo(b) == 0) to a sorted set that does not use an explicit comparator, the second add operation returns false (and the size of the sorted set does not increase) because a and b are equivalent from the sorted set's perspective.
So especially with SortedSet (and SortedMap) if the compareTo method returns 0, it assumes it as equal and doesn't add that element second time even the the equals method returns false, and causes confusion as specified in the SortedSet javadoc
Note that the ordering maintained by a sorted set (whether or not an explicit comparator is provided) must be consistent with equals if the sorted set is to correctly implement the Set interface. (See the Comparable interface or Comparator interface for a precise definition of consistent with equals.) This is so because the Set interface is defined in terms of the equals operation, but a sorted set performs all element comparisons using its compareTo (or compare) method, so two elements that are deemed equal by this method are, from the standpoint of the sorted set, equal. The behavior of a sorted set is well-defined even if its ordering is inconsistent with equals; it just fails to obey the general contract of the Set interface.