# Mathematical notation help.

Is saying !(6 = 4k) a right way to express that 6 is not divisible by 4? Or is there a better more accepted way? I'm writing a proof for discrete math and I need to be sure I'm doing it right.

• Using ! for "not" is more of a programming thing than a mathematics thing. – us2012 Oct 2 '13 at 20:44
• You can use \$\not|\$ = $\not|$ as a negated-divides symbol. – abiessu Oct 2 '13 at 20:45
• \nmid, producing $\nmid$, is widely considered nicer. – Daniel Fischer Oct 2 '13 at 20:46
• It may even be preferable to just express the thought in English rather than symbols if you are writing up a proof. – B. Mackey Oct 2 '13 at 20:49
• @DanielFischer Exactly! – dtldarek Oct 2 '13 at 20:50

If you don't want to use plain English (i.e. "6 is not divisible by 4"), then try one of

• $4 \nmid 6$,
• $6 \not\equiv 0 \pmod 4$,
• $6 \not\equiv_4 0$,
• $6 \bmod 4 \neq 0$,
• $6 \neq 4k$ for any $k \in \mathbb{Z}$.

I hope this helps $\ddot\smile$

• nitpick: the 4th one is also more a CS/programming thing than a math thing. – Dennis Meng Oct 2 '13 at 20:52
• @DennisMeng That's not true. You are right in the aspect that the binary operator $\bullet \bmod \bullet$ is more often seen in computer science. However, it is a valid mathematical notation and as such it is "a math thing" as any other mathematical notation. – dtldarek Oct 2 '13 at 20:57
• I guess it could also be something that differs based on experience; I remember that notation confusing the graders in my first discrete math class (where all the students in my recitation happened to be CS students), where my TA then quipped that "this really is the CS recitation" – Dennis Meng Oct 2 '13 at 20:59
• @DennisMeng Technically theoretical computer science is a subdomain of mathematics, so it doesn't really matter. Practically, one should use the most clear notation possible in the context, perhaps defining it if necessary (I can imagine places where \bmod would be preferred in math and where \pmod would be preferred in cs). I remember a huge confusion between me and my students regarding $\subset$, $\subseteq$ and $\subsetneq$. Since then I always try to disambiguate, for example in the case of \bmod I could take it in parentheses $(\bullet\bmod\bullet)$. – dtldarek Oct 2 '13 at 21:16

If you want to express "6 is not a multiple of 4" formally, you would write

$$4 \nmid 6$$ or

$$\not\exists k\in\mathbb{N}: 4k = 6$$

or even

$$\forall k\in\mathbb{N} : 4k \neq 6$$

The first one is obviously the most concise and easiest to understand. If you're not dealing with formal logic, you'd therefore go with $4 \nmid 6$.

Negations are usually writen as $\lnot$, so for example you could write $\lnot (4\mid 6)$ for $4\nmid 6$.