I am taking an Algebra class and i am relying too much on truth tables and Venn Diagrams to prove the equality or implication of operations between Sets. I understand that this is a negative habit because, firstly, in Venn Diagrams if there are more than 4 sets it is imposible to apply this method. Secondly, the truth tables, if there are more than 3 sets it can be really long and confusing, especially if i am working on paper, since the $ rows = 2^n $ with n being number of Sets.
For example to prove this equality I tried to practice the formal logic proof method but I failed and ended up doing a truth table. $ A − (B − C) = (A − B) ∪ (A ∩ C) $
Essentially I am asking for advices to prove via formal logic. Anything is useful, for example where to start or what to look first. Thanks in advance
Write out the basic theorems of in your case: set theory for example and try playing around with them.
Start out with easier examples so you get a hang of it, like how exactly you have to play around, there are certain ways which are followed which can be transformed into your hunch by seeing a bunch of simpler proofs. The more you see how things work, when tackling harder proofs, you'll know points from where to start.
In some cases, we start with taking an arbitrary element in sets required to prove and lead to RHS, using properties.
The list is not exclusive, feel free to contribute. Hope this helps.