Still sort of new to proofs and I'm not sure how to approach some of the questions. Both of these come from Tao's Analysis I and I have done all the exercises that don't involve iff statements and are just direct proofs about set equality so I think my problem stems from the fact I'm not sure how to deal with if then statements in the context of sets. I have read posts on MathSE about proving if then statements regarding sets but they didn't help me much because they didn't really explain why we do what we do.
Let $A$,$B$,$C$ be sets. Show that $A\cap B\subseteq A$ and $A\cap B\subseteq B$. Furthermore, show that $C\subseteq A$ and $C\subseteq B$ if and only if $C \subseteq A\cap B$.
For this first one I kind of came up with a proof. The first two statements are trivial since $A\cap B \implies A\land B$. Assume $C \subseteq A$ and $C \subseteq B$. Suppose $x\in C$ then by the assumption $x\in A$ and $x\in B \iff x\in A\cap B$ thus $C\subseteq A\cap B$. Now assume $C\subseteq A\cap B$ and suppose $x\in C$. Then $x\in A\cap B$ so $x\in A$ and $x\in B$ so $C\subseteq A$ and $C \subseteq B$.
If that above proof is right then it still feels weird because I didn't use the two statements I was asked to prove. I can see maybe a way to prove $C\subseteq A\cap B$ implies $C\subseteq A$ and $C\subseteq B$ but not the other way round. Let $x\in C$ and assume $C\subseteq A\cap B$. Then $x\in A\cap B$. So $x \in A\cap B \subseteq A$ and $x \in A\cap B \subseteq B$ thus $x\in A$ and $x\in B$ so $C\subseteq A$ and $C \subseteq B$.
For the second proof you have to show that $A\subseteq B$, $A\cup B=B$ and $A\cap B=A$ are all logically equivalent.
Let's say that I proved $A\subseteq B\iff A\cup B=B$ although it might be wrong. Now proving something like $A\cup B=B \iff A\cap B=A$ is where I struggle to formulate a proof.
Let's do $A\cup B=B \implies A\cap B=A$. Assume $A\cup B=B$. We want to show $A\cap B=A$. Now I should pick an arbitrary element $x$ right? But from which set? Or should I do it for both cases $x\in A\cap B$ and $x\in A$?
Maybe everything I've written here is wrong. That could very much happen because these proofs all just came from my inexperienced intuition of what a proof of this type should probably look like while I probably couldn't justify every step.
Best Answer
Your first proofs are correct. If I correctly understand the question, it is whether the word "Furthermore" means that the previous statements need to be used in proving the following one. I didn't read it that way, I think "furthermore" means the same thing as "also".
And yes, often proving an if and only if statement will be done by proving the if direction and the only if direction separately and combining them, as you have done.