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I need help writing a proof for a question from Velleman's "How to Prove It". The question is as follows:

Prove that for all real numbers x and y there is a real number z such that x + z = y - z

My attempt began by translating the goal into a $P \rightarrow Q $ statement.

$$\tag1 \forall x\forall y((x \land y) \in \Bbb R \rightarrow\exists z(x+z=y-z))$$

Assume x and y are some arbitrary real numbers.

Let$$\tag3 y-x = m$$

Then$$\tag4 \frac{y-x}{2}=\frac{m}2=z$$

Thus$$\tag5 2z= y-x =z+x = y-z$$

Because x and y were arbitrary real numbers $\tag1 \forall x\forall y((x \land y) \in \Bbb R \rightarrow\exists z(x+z=y-z))$Am I missing something or is this proof complete?

Timmy Chyrklund
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    Thanks for formatting in mathjax - note you need to surround the code with $ $ to actually make it math mode, and \commands need a space before the next letter (otherwise how would it know what's the end of the command word). – obscurans Jun 03 '20 at 21:51
  • Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Shaun Jun 03 '20 at 22:11
  • @Shaun I will keep that in mind for the future. I appreciate the reference and advice. – Timmy Chyrklund Jun 03 '20 at 22:12
  • In line 4 you can't conclude that $\frac m2 = z$, as $z$ has never been introduced. You can say. Let $z = \frac m2$ and then show that $x+ z = y -z$. – fleablood Jun 03 '20 at 23:26
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    In fact the shortest proof would be: For all $x,y$ then the number $z =\frac {y-x}2$ always exists. And $x + z = x+ \frac {y-x}2 = \frac {x+y}2$ and $y-z = y-\frac {y-x}2 = \frac {x+y}2$ so $x+z=y-z$. End of proof. – fleablood Jun 03 '20 at 23:31
  • I noticed I forgot to escape the $ $ itself on "how to turn things into mathjax" – obscurans Jun 04 '20 at 00:01

1 Answers1

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  1. The letter $m$ is used exactly once (for a line that does nothing)
  2. Presumably you don't actually want $\frac{z}{2}$ in line 4.
  3. With these changes, the proof looks fine.

After the changes:

  1. It's easier and more idiomatic to simply write $\forall x\in\mathbb{R},\forall y\in\mathbb{R},\exists z$. There doesn't need to be an explicit $\Rightarrow$ from the hypotheses to the conclusion in the statement.
  2. Instead of the $\rightarrow$, the logical connective is "such that", sometimes abbreviated s.t., sometimes $\mid$ as a symbol.
  3. Line 3, and the entire existence of $m$, can be removed without any effect as a simplification.
  4. You accidentally put another $=$ on line 5, the middle is $\Rightarrow$.
  5. The restatement at the end isn't particularly necessary, since we very much can see you're asserting the existence of $z$, and have just shown such a $z$.
  6. I would put QED right after line 5.

Nice work.

obscurans
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