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[DMDB] Intuition for FD and NF
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\subsubsection{Keys}
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\label{sec:relations-keys}
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\inlinedefinition[Candidate Key] The minimal set of fields that identify each tuple uniquely
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\inlinedefinition[Candidate Key] The minimal set of fields that identify each tuple uniquely (non-duplicate attributes (fields) typically)
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\inlinedefinition[Primary Key] A single candidate key, i.e. just a single field.
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We mark the primary key using \underline{underlining} in visual \glspl{schema}.
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Formally, we write $R(\underline{k : D_k}, a:D_a, b:D_b)$ for a given relation, with all valid instances fulfilling
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\[
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I \subseteq D_k \times D_a \times D_b \land \forall (k, a, b), (k', a', b') \in I : k = k' \rightarrow (a, b) = (a', b')
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I \subseteq D_k \times D_a \times D_b \land \forall (k, a, b), (k', a', b') \in I : k = k' \rightarrow (a, b) = (a', b')
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\]
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i.e. if the key is equal, so is the rest of the tuple (thus they are one and the same\footnote{\hlhref{https://www.youtube.com/watch?v=tGeSEbYzhBU}{Heidrun is her name}}).
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