So I'm reading through Zakon's Basic Concepts of Mathematics, and during the set theory introduction, we're asked to prove the basic operations of sets.
I'm coming directly from euclidean geometry, and I feel like this book does not present its definitions and axioms clearly enough. For example I have found out these proofs rely on basic laws of logic, e.g. the proof for the commutative law for sets is just the commutative law for logic, under a new name. That feels really silly. Is there a proof for this law in logic, or is it taken as an axiom?
I believe most of those logical laws are proven by writing out the truth tables for the statements to be compared and if they are identical, that indicates equality. For example, writing the tables for A^(B || C) and (A^B) || (A^C) would yield the same thing.
>>7941218
So the truth table for A v B would be:
A B AvB
T T T
T F T
F T T
F F T
whilst for BvA it would be
A B BvA
T T T
T F T
F T T
F F F
and this implies equality?
>>7941286
Sorry, last one there should be F for the the first table.
>>7941218
Truth tables are as much a proof as venn diagrams are.
The correct way to go about it uses propositional calculus (see: https://en.wikipedia.org/wiki/Propositional_calculus), and it has premises and inference rules.
>>7941310
So when reading a mathematical work, how do you know which logical system they're using? There are different logics, as I've come to learn.
Furthermore, assuming they're using propositional calculus, is the commutation law (within the rules of replacements) an axiom or is it somehow proved? See: https://en.wikipedia.org/wiki/List_of_rules_of_inference#Table:_Rules_of_Inference
>>7941310
Venn diagrams and truth tables are perfectly valid proof methods; they just aren't powerful enough to handle questions currently asked. What OP was describing was PC too, at an elementary level though.
>>7941310
Truth tables can be converted into that kind of proof easily, although you end up taking more steps than you need.
>>7941342
I think, outside of meta-mathematics and mathematical logic, people aren't too concerned with that.
From the wiki on RoR: "Within the context of a logical proof, logically equivalent expressions may replace each other. "
Which I think spells out that the commutative law, through the table, holds or is proven, in this case. Like you cautioned, commutation is not always present.
>>7941451
Yes, but that does not answer the question. Is the commutation law in PC an axiom? If not, how does the proof look, and what are the axioms?
>>7941464
The proof is from the truth table which shows given all valid inputs, the truth values of the two expressions are identical, and are therefore equivalent. Not entirely certain about the necessary axioms.
