Help with a set theory proof that $\bigcap\limits_{x\in[0,1]}[x,1]\times[0,x^2] =\{(1,0)\}$. I’m on a self teaching mathematical journey, and learning about proof writing. I’m desperate for constructive feedback on how to improve my skills. This is an exercise from The Book of Proof, which I am following.
$\bigcap\limits_{x\in[0,1]}[x,1]\times[0,x^2] =\{(1,0)\}$.
My Proof: $\forall x\in[0,1],1\in[x,1]$ and $0\in[0,x^2]$. This means $\forall x\in[0,1],(1,0)\in([x,1]\times[0,x^2])$, so $(1,0)\in\bigcap_{x\in[0,1]}[x,1]\times[0,x^2]$.
Now, we prove that no other element is in the intersection:
Let $(a,b)\in\bigcap_{x\in[0,1]}[x,1]\times[0,x^2]$. This means $(a,b)\in[x,1]\times[0,x^2], \forall x\in[0,1]$. In particular, if $x=0$,$(a,b)\in([0,1]\times[0,0])$ and if $x=1, (a,b)\in([1,1]\times[0,1])$. Then, $(a,b)\in([0,1]\times[0,0])\cap([1,1]\times[0,1])={(1,0)}$. Thus, $(a,b)=(1,0)$. QED
Is this correct? Any tips on how to make the proof better?
 A: You have the right idea, but towards the end, the notation is slightly wrong. You wrote

Then, $(a,b)\in([0,1]\times[0,0])\cap([1,1]\times[0,1])={(1,0)}$.

The correct way of saying it is

Then, $(a,b)\in([0,1]\times[0,0])\cap([1,1]\times[0,1])=\{(1,0)\}$.


Apart from that mistake (possibly a typo), it's fine. However, here are some general comments I have (me being nitpicky because you asked:). First, do not end a sentence with quantifiers; quantifiers belong at the beginning of a sentence. For instance you wrote

"This means $(a,b)\in[x,1]\times[0,x^2], \forall x\in[0,1]$."

It is clear to me (and anyone else who'll read this) as to what the sentence means, and there's no chance for misunderstanding. Also, in everyday language, that's also how we phrase sentences, and often in textbooks, you'll see it written this way with quantifiers at the end. However, when multiple quantifiers arise in a given proof, putting them at the end can get confusing. Placing quantifiers at the beginning of a sentence has the obvious advantage that it goes left-to-right, which is how we read and write in English, so there's no ambiguity. So, instead, write

This means, $\forall x\in [0,1],$ we have $(a,b)\in[x,1]\times[0,x^2]$.

Also, while we're on the subject of quantifiers, it is my opinion that unless you're studying (and writing assignments/papers on) logic exclusively, you should avoid those symbols (e.g. $\forall,\exists,\therefore,\because$ etc) as much as possible, and write things out in words. So, I would write the above sentence as:

This means that for every $x\in [0,1]$ , we have $(a,b)\in [x,1]\times [0,x^2]$.

Sure, the sentence looks longer, but I think it's easier to read.
My next comment sort of follows the previous one, and that is to avoid excessive notation, because it makes things hard to read (we are humans, not robots:) and we're more likely to make mistakes (e.g. the type error I mentioned in the beginning). So, if it were me, I would rephrase your second paragraph as follows:

Let $(a,b)\in\bigcap\limits_{x\in[0,1]}[x,1]\times[0,x^2]$. This means for each $x\in [0,1]$, we have $(a,b)\in[x,1]\times[0,x^2]$. In particular, setting $x=0$, we get $(a,b)\in[0,1]\times[0,0]$, and thus $b=0$. Likewise, taking $x=1$, we see that $(a,b)\in[1,1]\times[0,1]$ and hence $a=1$. Putting these two results together, we have $(a,b)=(1,0)$.


But like I said, I'm just being super nitpicky because you asked. Your proof (apart from the minor error) is great: the logic behind your argument is clear and it's clearly written. For some general remarks, take a look at John Lee's remarks on writing proofs; there's lots of good stuff there.
