Question from Coursera's Introduction to Mathematical Thinking:
Prove that if 5 investors split a payout of \$2,000,000 at least 1 investor receives at least $400,000.
My proof :
let i : investor
let e : equal amount
let p : payout
let s : split
Set of 5 investors : $s : \{i1,i2,i3,i4,i5\}$
p(e) : 2M / 5 = $400'000
As each investor receives an equal payment then at least 1 investor receives 400'000, in other words : $\forall(s)[e(i)] => p(400'000)$
Is this a valid proof?
If each investor gets less than $\$400,000$, then the sum of their payments is less than $\$2,000,000$, a contradiction; thus at least one investor gets at least $\$400,000$.
No, your proof uses the assumption that each investor gets equal amount which is not given.
It can be shown using reductio ad absurdum relatively easy:
If you don't assume that the investors gets a payout of $p_i$. This means that the total payout $p=\sum p_i$. Now assume that the statement is false, that means the negation: all investors get less than \$400,000 ($p/5$). Now we have $p_i < p/5$ so $\sum_{j=1}^5 p_j < \sum_{j=1}^5 p/5 = p$ which contradicts that $\sum p_j = p$. Therefore we can conclude the assumption is false and the statement to be proven is true.
There are several problems here.
let i : investor
I'm not sure what the colon is supposed to mean here. What are you trying to communicate here, in plain English?
let e : equal amount
You should be more specific than "equal amount". By "equal amount", do you mean "the amount each investor receives if the payout is split equally"? If so, write that out explicitly:
Let $e$ be the amount each investor receives if the payout is split equally.
let p : payout
Again, you should be more specific than "payout". Do you mean "the amount of money paid to each investor"?
Set of 5 investors : $s : \{i1,i2,i3,i4,i5\}$
It looks like you're giving the letter $s$ a second meaning here. You defined it as meaning "split" earlier, and now you're defining it as being the set of investors. If you have two different meanings, you need to use two different letters.
p(e) : 2M / 5 = $400'000
You haven't defined what the notation $p(e)$ means. Perhaps it would have been better to write that part out in prose instead of using notation:
If the payout is split equally, the amount received by each investor is $\$2,000,000 / 5 = \$400,000$.
As each investor receives an equal payment [...]
Hold on. We haven't been told that each investor receives an equal payment, so a proof that relies on this assumption is incorrect. Can you find a different proof which doesn't rely on this assumption?
[...] then at least 1 investor receives 400'000, in other words : $\forall(s)[e(i)] => p(400'000)$
Your notation doesn't make sense if I try to translate it back into prose. This reads as:
Suppose that for all $s$, the equal amount of the investor. Then payout of $400,000.
Write this out in plain English instead of notation, and then maybe we can find a way to convert it into clear notation.
If $$ \forall i \in \{1,\dotsc,5\}: p_i < 400\,000 \quad (*) $$ then $$ S = \sum_{i=1}^5 p_i < \sum_{i=1}^{5} 400\,000 = 400\,000 \sum_{i=1}^{5} 1 = 400\,000\cdot 5 = 2\,000\,000 $$ but the sum $S$ needs to be $2\,000\,000$.
So $(*)$ is false, the negation is $$ \exists i \in \{1,\dotsc,5\}: p_i \ge 400\,000 $$