Many authors define group as a non empty set with the binary operation $*$ with the following 4 properties:
- closure,
- associative,
- identity,
- inverse.
My question is that it is obvious that a set with a binary operation is always closed, then why they mention closure property?
Best Answer
Although it is redundant in the case you are mentioning, in practice people talk about binary operations without mentioning exactly what set they are on. For example it makes sense to add irrational numbers, although technically addition doesn't form a binary operation on the irrational numbers because the sum of two irrational numbers may be rational. So here is where the concept of closure becomes important. People don't actually say "addition does not give a binary operation on the irrational numbers", they say "the irrational numbers are not closed under addition". So it is worth emphasizing the concept of closure and that it is part of the definition of group so that it is not "lost" in the technical definition of binary operation, which does not completely coincide with the way people actually speak.
See edits more details if you'd like.