Why Projection Operator in relational algebra eliminates duplicates?

Question

What are the consequences of eliminating duplicates in Projection Operator?And yes, why there is no duplicate elimination in real systems, unless specified?

Answer 1

If something is true, then saying it twice won't make it any truer.

The required duplicate elimination was left out of the early implementations of "relational" systems because the engineers were afraid of the performance implications. And even when keywords like DISTINCT were newly introduced / newly supported in an existing system, the algorithms used for it were typically so naive that "performance implications" appeared to be very real and/or inevitable. Things have remained like that ever since.