Question

To prove an \(n\) - ary relation with a primary key defines a function.

Short answer

It is proved that an \(n\) - ary relation with a primary key defines a function.

Step-by-step solution

Given data

An \(n\) - ary relation \(R\) with a primary key. In other words, a collection of \(n - \) tuples, where, say the first tuple uniquely determines the other \((n - 1)\) tuples.

Definition of \(n\) - ary relation

By definition, a function\(f\)from a set\(A\)to a set\(B\)is an assignment which assigns to every element\(x\)of\(A\)a unique element of the set\(B\).

Proof of \(n\) - ary relation defines function

As per given data, the relation \(R\) is a set of \(n - \) tuples.

\(R = \left\{ {\left( {x,{y_1},{y_2}, \ldots ,{y_n}} \right)} \right\}\)

Now \(R\) has the set \(X\) of first coordinates (domain) as a primary key. Thus every \(x\) determines a unique row of \((n - 1)\) tuples. This means that the assignment is \(g:x \to \left( {{y_1},{y_2}, \ldots .,{y_{n - 1}}} \right)\) where \((x \in X)\).

Is a well-defined function from the set \(X\) to the set of other \((n - 1)\) tuples.