TITLE: Precise definition of free group
QUESTION [9 upvotes]: I have seen the definition of a free group go like this:

Let $S = \{s_i : i\in \mathbb{N} \}$ be a countable set. Let $S^{-1}$ be the set $\{s_i^{-1}: i\in \mathbb{N}\}$. Here one is to understand $s_i^{-1}$ as a notation for an element of this other set. Let $W(S)$ be the set of words in $S$ and $S^{-1}$. That is, the elements in $W(S)$ are strings of elements from $S$ and $S^{-1}$. One then defines a function $W(S) \times W(S) \to W(S)$. One makes $W(S)$ into a group (free group) be defining an operation with certain cancellation operations (so $s_is_i^{-1}$ is the empty word).

All of this makes much sense intuitively. My question is: 
Is there is a more precise way to define this?
What I mean is, talking about strings, empty word, and cancellation properties doesn't seem very clear (precise). I understand that a more precise definition would be harder to grasp intuitively, but I was just wondering if there, for example, was some extremely precise way to define a free group only relying on language from, for example, set theory.
Edit: I am in beginning Abstract Algebra.

REPLY [2 votes]: In my opinion, precise definition of a free group should be based on the concepts word, sub-word, empty word, inverse word, cancelation, reduced word, and juxtaposition. These concepts are easy to grasp and is really clear and precise. 
Even if you define a free group in terms of category theory (functors and homomorphism), known as a universal property, you would want to apply to the notions of words and generators in order to prove the existence of a free group. Of course, there may be purely categorical proof of existence of a free group (like here), but it is not simpler than usual proof using words and generators. This is a good resource explaining this approach.
Another reason for giving a definition of a free group in terms of words and generators is that it is easier to grasp for computer scientists who usually deal with strings, algorithms or rewrite systems.