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I am studying database functional dependency. It says if A can functionally determine all other attributes, it is considered to be a key.

What if 2 tuples are exactly the same, can we still say A is a key? To me A cannot determine an unique tuple.

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Pragmatically, You are correct that when you declare something as a primary key, that it must correspond to one tuple. Databases will not allow you to store a tuple in a table if the primary key already exists for a tuple in the table.

Theoretically, If $R$ is a relation with the following set A of attributes $\{a_1, a_2,..., a_n\}$, then a primary key on the relation $R$ would be a subset $P \subset A$ such that for $F:P \rightarrow A$, we have $\forall x,y \in P $ $F(x)=F(y) \rightarrow x=y$. This basically just means that the primary key forms a one-to-one mapping from the domain of keys to the range of corresponding tuples.

If you have that two tuples are the same $A$ and $B$, then this implies that $A = B$,and we still have that $F(A)=F(B)$ so that there is no violation, in that sense.

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"What if 2 tuples are exactly the same ?"

If you are using relational database model, all tuples (rows) are unique. No two tuples(rows) can be same in a relation (table).

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