Minimal cover FD - database-normalization

Question

I have the following exercise text:
Stack Overflow is a community where users (US), with a specific number of years of experience (Ye), can ask questions and give answers. Based on the given answers, each user earns a reputation (Re) and a badge (Ba), a sort of medal (e.g., bronze, silver and gold badge). The number of years of experience starts from the registration date (Da) on Stack Overflow. Each user can gain some money (Mo) by answering a question. Please note that the description above is not referring to the real Stack Overflow, but it’s useful to define the following exercise. The following FDs are satisfied:
Each user has a specific number of years of experience and a specific badge
The years of experience uniquely determine the user’s reputation and the amount of money for each answer
For each user and for each reputation is defined a unique badge
The registration date determines a single value of years of experience
Each user earns a specific amount of money for each answer
The FDs I defined are the following ones:
YeBa -> US
Ye -> ReMo
Ba -> USRe
Da -> Ye
Mo -> US
I'm not sure about the third one and the last one. For the third one I don't know if it is correct the one I wrote, or there must be two FDs (Ba -> US, Ba -> Re). For the last one I'm not sure how to represent it, because the sentence doesn't tell exactly what I wrote, and I think it is wrong because it asks me to apply the minimal cover and with these FDs it is not possible.

Answer 1

A functional dependency X → Y expresses the fact that values of the set of attributes Y are determined in a unique way from the values of the set of attributes X. In other words, for each combination of X values there is a unique, specific combination of values of Y.
Let's examine the facts described in the exercise.
Each user has a specific number of years of experience and a specific badge
You have represented this fact with the dependency:
Ye Ba → US
but this is exactly the opposite. With the above FD, in fact, you are saying that given a number of years of experience and a certain badge, there is only a user with such years and badge, but in the reality it is possible that different users have the same years of experience and badge. The fact specified by the sentence can instead be expressed by the FD:
1. US → Ye Ba
that is, each user determines (has uniquely) a number of years of experience and a badge.
Let see the other facts:
The years of experience uniquely determine the user’s reputation and the amount of money for each answer
2. Ye → Re Mo
(this is correct)
For each user and for each reputation is defined a unique badge
This is somewhat ambiguous, as pointed in a comment, in the sense that the badge is probably determined by the reputation only, but if we follow the specification and write the FD corresponding to it, we can write:
3. US Re → Ba
(not the viceversa), and this will be simplified when we will compute a minimal cover.
The registration date determines a single value of years of experience
4. Re → Ye
(correct)
Each user earns a specific amount of money for each answer
5. US → Mo
again, there is a certain, specific, amount of money earned by a user (and not the viceversa, since that would mean that for a certain amount of money earned, only a single user can earn it).
So we have now the dependencies:
US → Ye Ba
Ye → Re Mo
US Re → Ba
Re → Ye
US → Mo
and we can compute a minimal cover from it. Here is a possible computation (note that it is always possible to compute a minimal cover from any set of dependencies).
First we transform the dependencies with a single attribute on the right:
Re US → Ba
Re → Ye
US → Ba
US → Ye
US → Mo
Ye → Mo
Ye → Re
In Re US → Ba, Re is extraneous since {US}+ = (Ba Mo Re US Ye). So we replace this dependency with US → Ba (which is already present).
From the remaining dependencies, we can now remove the redundant dependency:
US → Mo
This is because we have US → Ye and Ye → Mo.
So, the final set of dependencies (a minimal cover) is:
Re → Ye
US → Ba
US → Ye
Ye → Mo
Ye → Re