Proof writing advice.

I am new to writing informal proofs. I am fine with formal proofs, but the transition is jarring. I do understand the necessity of informal proofs. My questions are:

1. How do I figure out which steps to include?

2. How should I format the proofs? Most informal proofs are presented in a paragraph form. Consider instead the proof format below. Is this acceptable? Professors, what would you say to a student who wrote a proof like this? I find this easier to write than paragraphs, since no effort is spent on choice of words.

Prove: $\bigcap \{ \mathcal{P}(X) : X \in A \} = \mathcal{P}(\bigcap A)$.

1. First we prove that $\bigcap \{ \mathcal{P}(X) : X \in A \} \subseteq \mathcal{P}(\bigcap A)$.

1. Assume $y \in \bigcap \{ \mathcal{P}(X) : X \in A \}$.
2. Then, for all $X \in A$, $y \in \mathcal{P}( X)$.
3. Then, for all $X \in A$, $y \subseteq X$.
4. Assume $z \in y$.
5. Then, for all $X \in A$, $z \in X$.
6. Then, $z \in \bigcap A$.
7. Then, for all $z \in y$, $z \in \bigcap A$.
8. Then, $y \subseteq \bigcap A$.
9. Then $y \in \mathcal{P}( \bigcap A )$.
2. Now we prove that $\mathcal{P}(\bigcap A) \subseteq \bigcap \{ \mathcal{P}(X) : X \in A \}$.

1. Assume $y \in \mathcal{P}(\bigcap A)$.
2. Then, $y \subseteq \bigcap A$.
3. Assume $z \in y$.
4. Then, $z \in \bigcap A$.
5. Then, for all $X \in A$, $z \in X$.
6. Then, for all $X \in A$ and for all $z \in y$, $z \in X$.
7. Then, for all $X \in A$, $y \subseteq X$.
8. Then, for all $X \in A$, $y \in \mathcal{P}( X )$.
9. Then, $y \in \bigcap \{ \mathcal{P}(X) : X \in A \}$.
3. Thus, $\bigcap \{ \mathcal{P}(X) : X \in A \} = \mathcal{P}(\bigcap A)$.

• That is precisely how I write proofs in my personal notes. I find it easier to follow everything, and to spot errors. I also include a precise list of all the hypotheses. In my case, I would've included something like: ### Hypotheses 1. $A$ is a set. I once showed my notes to a Professor and he didn't like it, though. – étale-cohomology Nov 22 '16 at 3:46

2 Answers

I wouldn't be so structured. Being formal is fine, until it makes communicating a proof hard. So, you want to show that $$\bigcap_{X\in A} P(X) = P(\cap A)$$

The left hand side consists of those sets $z$ that are subsets of every set in $A$. Because $\cap A$ is the intersection of all sets in $A$, and because $z$ is a subset of all of the sets in $A$, $z$ is a subset of $\cap A$. Conversely, assume that $z$ is a subset of $\cap A$. This means that $z$ is contained in every set in $A$. That is, $z$ is a subset of every $X\in A$, which means $z$ is in $\bigcap_{X\in A} P(X)$.

• Thanks for the answer. I was waiting for other answers but I guess they're not coming :( What I learned by comparing your answer with mine is that informal proof is not just formal proof with missing steps. It should communicate intuition behind the proof, which could require entirely different wording than a proof that skips obvious details but otherwise follows a formal proof in structure (like mine). Is this correct? If you have any other suggestions, I'd be glad to know. – stranger Nov 28 '16 at 8:53