How to go about proving the basic operators in relational algebra are independent of each other? 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.
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.