关于陶哲轩《分析I》命题3.1.27中两个集合论命题证明有效性的问询

Great question—let’s break down why both of your proofs are actually solid, even if they feel a bit "too straightforward" at first glance.

First Proof: $A \cap (X \setminus A) = \emptyset$ #

Your double-inclusion approach is exactly the right way to prove set equality, and every step checks out:

• For the $\emptyset \subseteq A \cap (X \setminus A)$ direction: Vacuous truth works here because there are no elements in $\emptyset$ to contradict the implication. This is a standard, accepted way to handle subset claims involving the empty set.
• For the $A \cap (X \setminus A) \subseteq \emptyset$ direction: By assuming $x \in A \cap (X \setminus A)$, you derive the contradiction $x \in A \land x \notin A$. In logic (and set theory), any statement that leads to a contradiction implies the conclusion you want (here, $x \in \emptyset$)—since contradictions can’t be true, the only way the implication holds is if there are no such $x$, which is exactly what it means for the set to be empty. This is a rigorous way to prove a set is empty.

Second Proof: Associativity of Intersection #

Your concern about "using associativity of logic to prove associativity of intersection" is totally understandable, but this is not a non-proof—it’s actually the correct way to do it, given how set intersection is defined.

In Tao’s book (and most axiomatic set theory), the intersection operation is defined directly using logical conjunction: $x \in A \cap B$ if and only if $x \in A \land x \in B$. So when you expand the membership statements for $(A \cap B) \cap C$ and $A \cap (B \cap C)$, you’re just unpacking the definitions into logical statements.

The associativity of $\land$ (logical "and") is a fundamental principle of propositional logic—either it’s taken as an axiom or proven early on. Since set equality reduces to equivalence of membership conditions, using the associativity of $\land$ to link the two membership conditions is completely rigorous. You’re not "begging the question" here; you’re using the foundational definitions that connect set theory to logic.

The reason it feels trivial is because it is straightforward once you make that connection—but that’s a good thing! It means you’re correctly applying the definitions instead of overcomplicating things. Tao likely expects this kind of proof because it reinforces the link between set operations and their logical counterparts.

In short: Both of your proofs are valid, rigorous, and align with how these results are typically proven in introductory analysis/set theory.

备注：内容来源于stack exchange，提问作者Eric