I just started my intro class to real anylysis, and something is bothering me and I thing I need some clarification because I really want to understand this because first I love it and it seems like it will partially awnser most of the questions I've been asking myself about all previous classes.
The first chapter is dedicated to an accurate definition of the real number system.
We first defined the axioms on the addition and the multiplication that made the real number set a Field.
So from what I understand, a Field is any (non-empty?) set F such that we define two binary opearations "addition and multiplication" that must hold under the defined axiomes (would it be true to say that we could define ANY operations as long as they verify the axioms?)
- Closure under addition and multiplication (could we say intuitively thatapplying those operations on any arbitrary element of the set $S$ does not "expel" the result out of the set?)
- Additive and multiplicative commutativity/associativity (i.e the order on which we apply those operations does not matter)
- The existence of an Identity for addition and multiplication
- The existence of an inverse
We then say that an order on a set $S$ is a relation between its elements such that only one of the statements holds for every $x,y\in S$
$x=y$ or (exclusive) $x > y$ or $x < y$
and we say that $S$ is an ordered set if $x>y$ implies $x - y > 0$ etc.
So from that my questions would be:
How do we define an operation? I don't know how to "visualize it". Is it like a transformation that takes two objects of a set and gives a new one (possibly the same)
And does $0 $ being kind of the "limit between negative and positive numbers" is a consequence of some previous axiom or simply an arbitrary decision just so we could define the order on a set.
I'm sorry my question is kind of large but I would be very grateful if someone could maybe clarify those ideas or maybe point me in the right direction.
Thank you!