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The proof of the statement below is a homework question, however I did not tag this question as such since I don't need the actual proof: I have already proved the statement wring and don't need a solution; my question here is strictly in regards to the mathematical notation I am using to represent a statement.

Consider the following statement:

"If $a$ and $b$ are integers, then there are integers $m$ and $n$, such that $a=m+n$ and $b=m-n$."

So my question is, will the following be a correct representation of the statement above:

$$(\forall a\in\mathbb{\mathbb{Z}},\forall b\in\mathbb{Z})(\exists m\in\mathbb{Z},\exists n\in\mathbb{Z}: a=m+n, b=m-n)$$

Is this a clear representation or is there a more common way of writing such statement that I don't know about?

Thank you!

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Yes, the notation is fine. One slight improvement would be $a=m+n\land b=m-n$.

However you need to understand that mathematical content is simpler when presented in words.

The statement you wrote:

If $a$ and $b$ are integers, then there are integers $m$ and $n$, such that $a=M=n$ and $b=m-n$.

If very clear. On the other hand when parsing (in my head) something of the form:

$$(\forall a\in\mathbb{\mathbb{Z}},\forall b\in\mathbb{Z})(\exists m\in\mathbb{Z},\exists n\in\mathbb{Z}: a=m+n\land b=m-n)$$

I have to step carefully and verify that what is written there makes sense, the result in my mind is exactly the original textual statement. So while it seems "cool" to write things like that (I used to think that in my freshman year) it does not make things clearer. Especially if the original text which you want to translate was very clear to begin with.

To top this off, I suggest you to go to the library and pick a book, or even journal, and see that most of the statements are written in words rather than mathematical notation. This is simply clearer.

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    $\begingroup$ "and" is even better than $\land$. $\endgroup$
    – JeffE
    Jun 9, 2012 at 16:33
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    $\begingroup$ @JeffE: $\land$ is an improvement over $,$ (a comma). At least in my neck of the woods. $\endgroup$
    – Asaf Karagila ♦
    Jun 9, 2012 at 16:33
  • $\begingroup$ @Asaf Karagila: actually, I absolutely agree with you. I always end up spelling out (at least in my mind) mathematical statements in English, since condensed notation often confuses me; this got me fine through most of my university studies. But! I am taking this first year course (you guessed right) and the chapter from the textbook is on logic/notation, and authors make a point of rewriting English statements in mathematical notation, so I decided to follow the suit. Thanks! $\endgroup$
    – 0x4B1D
    Jun 9, 2012 at 16:35

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