What are these set operations, and why do they give different results?

The set operations you have mentioned are:

^ - symmetric difference (XOR):

Return a new set with elements in either the set or other but not both.

Example: {'1', '2', '3'} ^ {'2', '3', '4'} = {'1', '4'}

| - union (OR):

Return a new set with elements from the set and all others.

Example: {'1', '2', '3'} | {'2', '3', '4'} = {'1', '2', '3', '4'}

There are also other set operations in python:

& - intersection (AND):

Return a new set with elements common to the set and all others.

Example: {'1', '2', '3'} & {'2', '3', '4'} = {'2', '3'}

- - difference:

Return a new set with elements in the set that are not in the others.

Example: {'1', '2', '3'} - {'2', '3', '4'} = {'1'}

The order of precedence for these operations is -, &, ^, |, so in your example, we first apply ^:

>>> y^z{'a', 'c', 'e', 'f', 'g', 'h', 'i'}

And then |:

>>> x|{'a', 'c', 'e', 'f', 'g', 'h', 'i'}{'a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i'}

The different outputs you describe are actually the same set, as sets are not ordered.

>>> {'c', 'h', 'f', 'd', 'b', 'i', 'g', 'a', 'e'} == {'a', 'd', 'h', 'f', 'b', 'g', 'e', 'c', 'i'}True

Any order shown in the string representation of a set is an implementation detail and should not be relied upon as it will vary unpredictably, as you have found.