We will make no such distinction. A set may contain the same elements as another set. The universal set is chosen arbitrarily, but must be large enough to include all elements of all sets under discussion. During the 19th century, mathematicians were able to tame such paradoxes and about the turn of the 20th century Whitehead and Russell started an ambitious project to carefully codify mathematics. Set theory was developed about this time and serves to unify the many branches of mathematics.
Geometry has a long history of such systematic study.
The ancient Greek Euclid similarily codified the mathematics of his time into 13 books called The Elements . Although these books were not limited to Geometry, that is what they are best known for. The Elements carefully separated the assumptions and definitions from what was to be proved. The concept of proof dates back another couple hundred years to the ancient Greek Pythagoras and his school, the Pythagorean School. Once we have created the concept of a set, we can manipulate sets in useful ways termed set operations .