your question is a bit misstated. you are asking about trut…

your question is a bit misstated. you are asking about truth conditions for compound sentences, which is a semantic matter, not a proof-theoretic one (proofs from axioms to theorems consist of syntactic steps governed by rules of inference which do not involve notions such as truth).

you can state the truth conditions for classical conjunction with a simple clause:

'A&B' is true if and only if 'A' is true and 'B' is true.

that settles the truth conditions, although the clause presupposes you understand the meaning of the English words 'if and only if' and 'and' which are used to state it (which you do).

as for the proof theory corresponding to such a semantics, the number of axioms you need to govern classical conjunction depends on how you set up the logical system: you could have different systems that are equivalent (i.e. which yield exactly the same theorems) but have different axioms. you can even set up a logical system with zero axioms (but many rules of inference, this is done in what's known as natural deduction). speaking broadly though, you would need an axiom governing what you can deduce from a conjunction, and another axiom governing what premises you need in order to deduce a conjunction.

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no AI was used for these answers, i find the responses you got above to be quite misleading.