Do I need to prove both directions of this if and only if statement for sets?

Question

I need to show that $S{1} = S{2}$ iff $$(S1 \cap \bar{S2}) \cup(\bar{S1} \cap S2) = \emptyset$$

Ok So I'll show that

$1.$ if $S{1} = S{2}$ then $(S1 \cap \bar{S2}) \cup(\bar{S1} \cap S2) = \emptyset$

$2.$ if $(S1 \cap \bar{S2}) \cup(\bar{S1} \cap S2) = \emptyset$ then $S{1} = S{2}$

Proof of 1 is straightforward where I assume $S1=S2$ and then I prove that $(S1 \cap \bar{S2}) = \emptyset $ and $(\bar{S1} \cap S2) = \emptyset$ , since $\emptyset \cup \emptyset = \emptyset$. For each, $(S1 \cap \bar{S2})$ and $(\bar{S1} \cap S2)$ , I use $S = S1 = S2$ and suppose there IS an element x that belongs to each and contradiction arises because x belongs one but doesn't belong to the other. So this proof is complete

However, do I need to also prove statement 2?

I'd prove it by contradiction, so assuming $S1 \neq S2$. So then there $\exists$ $x \in S1$ but $x \notin S2$ or vice versa. And then we arrive at contradiction. Since x belongs to $S1$ and x belongs to $\bar{S2}$ , and we KNOW that $(S1 \cap \bar{S2}) = \emptyset$ ,so there's a contradiction and so $S1=S2$. I do this for the "vice versa" step also.

Is this proof correct? Do I always need to prove iff statements by proving both sides? Thanks anyone who helps.

Answer 1

Do I always need to prove iff statements by proving both sides?

Yes. $A \iff B$ is really just a compact way of saying two different statements imply each other. Of course, that needs to be justified at some point. Typically that means showing $A \implies B$ and $B \implies A$. There are exceptions to this rule depending on the circumstances and how you choose to go about proving things but that's something you'll learn more in detail as you continue learning.

(And even with the exceptions, that's still proving it both ways, it just circumvents some of the work. You'd basically get $A \iff B$ right off, instead of having to prove both ways individually. Sadly a concrete example of what I'm trying to get at escapes me for the time being.)

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Is this proof correct?

Well, this presumably isn't the proof you're submitting for your class (or wherever this is coming up), since you seem to be just hitting the high points. So as-is, the proof isn't correct, but that's me being rather pedantic.

At least in terms of the overall process, your proof seems fine.

Answer 2

The proof is correct. Iff means if and only if and so by definition you must prove both to prove an iff statement.