Proof writing advice.

Question

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.

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

Answer 1

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) $.

Answer 2

See whether this link helps you: http://mypages.valdosta.edu/plmoch/MATH4161/Spring%202004/informal_proofs.htm. This is also very useful : https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://people.cs.pitt.edu/~milos/courses/cs441/lectures/Class6.pdf&ved=0ahUKEwiAqP_W17nQAhVLNY8KHfCmCHEQFggbMAA&usg=AFQjCNGZJxYFtzDjN6b3CCNXP3smeFQwOw&sig2=zUfeNvUxhGHr6zstsaKVIw.