Prove $A\cap (B\cup C) \subseteq (A\cap B) \cup C$

Solution 1:

First of all, I would start with saying "let $x\in A\cap(B\cup C)$". Your line 1 follows easily, but it's important that your proof follow the form $x\in X\to x\in Y$ to show that $X\subseteq Y$.

Next, you have the right idea for line 2, but you've already defined $x$. Get rid of the "let" in that statement, and replace it with something like "it must be the case that". You're showing a logical step, not a new definition.

Now you have a disjunction (aka OR statment), and you can finish the proof by cases. You know $x\in A$. Show that if $x\in B$, then $x\in(A\cap B)\to x\in(A\cap B) \cup C$. Show that if $x\in C$, then $x\in(A\cap B) \cup C$.

Solution 2:

Short answer: Case distinction.

A bit longer answer: In your second line you notice that one of two things is true: You might have $x \in A$ and $x \in B$ or you might have $x \in A$ and $x \in C$ (or you might have both). Since you don't know which of the two is the case for your generic $x$, you have to show that your desired conclusion $x \in (A \cap B) \cup C$ follows in both cases.

Formally, this is done by a case distinction, i.e.:

Case 1: If $x \in A$ and $x \in B$, then ... intermediate steps ... which shows $x \in (A \cap B) \cup C$

Case 2: If $x \in A$ and $x \in C$, then ... (different) intermediate steps ... which shows $x \in (A \cap B) \cup C$

An additional, stylistic, remark: In mathematical proofs "let" is used for a very specific purpose -- namely, introducing new objects. So, using "let" in your first line of thought is correct. In your second line, however, you shouldn't use "let" since you aren't introducing a new $x$ but rather deducing a fact about the x you introduced prior. So, I would just say "Then $x \in A$ ..." here.