[Math] Show that if $A$ and $B$ are sets, then $A \subseteq B$ if and only if $A \cap B = A$.

elementary-set-theory

I'm working through a real analysis textbook, and it starts out with set theory. The first exercise is

Show that if $A$ and $B$ are sets, then $A \subseteq B$ if and only if $A \cap B = A$.

I think I proved it correctly but I'm not sure. Here's what I did. I proved that if $A \subseteq B$, then $A \cap B = A$ the same way as this answer did (https://math.stackexchange.com/a/446114/93114), but I want to make sure I proved the converse correctly because it seems really easy (yes it's the first problem in the book, but still) and math usually isn't this easy for me, even the basic stuff!

Proof of "If $A \cap B = A$, then $A \subseteq B$."

If $x \in A \cap B$, then $x \in A$ and $x \in B$, but this applies to all $x \in A$ because $A \cap B = A$. So, for any $x \in A$, we know that $x \in B$, so $A \subseteq B$.

Am I on the right track?

Best Answer

Since you are just starting, I would suggest to be verbose instead of pulling everything in a single sentence.

To prove $A \subseteq B$ iff $A \cap B = A$, you have to

  1. show $A \cap B = A$ given $A \subseteq B$. That is to
    1. show $A \cap B \subseteq A$.
    2. show $A \subseteq A \cap B$.
  2. show $A \subseteq B$ given $A \cap B = A$.

Proof:

1.1) It is trivially true. You don't need to be given $A \subseteq B$ for it to be true.

1.2) If $x \in A$, then $x \in B$ since we are given $A \subseteq B$. Then $x \in A$ and $x \in B$ are both true. Therefore, $x \in A \cap B$.

2) From $A \cap B = A$, we know $x \in A$ and $x \in B$ whenever $x \in A$. If $x \in A$, then it must be the case that $x \in B$. Therefore, $A \subseteq B$.

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