The five basic operator select, project, cross, union and diff in relational algebra are independent of each other. I'm trying to formally prove this statement but can only progress for cross product as it would have columns greater than the rest which cannot be made possible by others. But can't think of anything for Select etc.
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1 Answer
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Here are some hints:
- Select: consider the relation $\{(1),(2)\}$.
- Project: number of columns can't decrease.
- Cross product: number of columns can't increase.
- Union: consider the database $\{(1)\},\{(2)\}$ with identical attributes.
- Difference: similar to union.
answered Sep 3, 2012 at 5:13
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$\begingroup$ Can you elaborate more about the relation {(1),(2)} part, could not understand that, is it possible to prove this mathematically? $\endgroup$– gizgokCommented Sep 3, 2012 at 6:39
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$\begingroup$ Use mathematical induction. The point is that you can't get $\{(1)\}$ or $\{(2)\}$ without using select. $\endgroup$ Commented Sep 3, 2012 at 14:19
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$\begingroup$ Can't {(1)} be obtained via difference of {(1),(2)} and {(2),(3)}? How would one guarantee {(2),(3)} hasn't been obtained somehow at prior induction step? $\endgroup$ Commented Sep 4, 2012 at 18:00
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$\begingroup$ @Tegiri You're right that $\{(1)\}$ can be obtained via the difference of $\{(1),(2)\}$ and $\{(2),(3)\}$. Fortunately, there is no row $(3)$ anywhere in the database. $\endgroup$ Commented Sep 4, 2012 at 21:15
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1$\begingroup$ @Tegiri Regarding your first point, perhaps one can argue that selection is join, but the formal definition is different. Regarding your second point, you're right, it does look more complicated, since all tables can be produced using the other four operations. Presumably some functions cannot be realized without set difference. $\endgroup$ Commented Sep 4, 2012 at 22:36