I've been reading about the relational algebra equivalence rules as we see them Here or Here and also I have been looking for proofs of those rules everywhere. I see there are some ways such as truth table or diagrams to show the equalities but I was wondering if anyone's aware of a robust mathematical way to prove those rule equalities?
I started reading and it led me to the De Morgan forth paper "On the logic of relations, 1860" as he says
we must know and use the properties * A is B gives B is A' and ' A is B and B is C, compounded, give A is C *
which is an understandable basic logic but I'm just wondering if there is any way to prove those equivalence rules by mathematics?
I appreciate any help here.
The truth-table method is a perfectly robust method to do this. The equivalence of two statements in propositional logic means that in every scenario, the two statements have the same truth-value. Since truth-tables explore all relevant types of scenarios, they do exactly what needs to be done to demonstrate equivalence.