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If I want to say "$n$ is greater than or equal to $k$," I can say $n \geq k$.
If I want to say "$n$ is not greater than or equal to $k$," I can say $n \ngeq k$.

What if I want to say "$n$ is greater than and not equal to $k$?" (In other words, $n > k$ and $n \neq k$). If I'm stating that "$n$ is greater than and not equal to $k$," might it be better to just state it as two separate conditions rather than using one symbol (if one exists).

Asaf Karagila
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supersmarty1234
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    If $n>k$ then $n\neq k$... – Andrew Li Apr 16 '18 at 20:20
  • Obviously. But what I'm looking for is dependent on the specific context. One example may be if I'm trying to prove that $n \geq k$, If I find this that this statement is false, I may want to say $n \ngeq k$ rather than $n < k$. – supersmarty1234 Apr 16 '18 at 20:23
  • Note that for non-total orders, $n \ngeq k$ is not equivalent to $n < k$. However, $n > k$ always implies $n \neq k$. – Magdiragdag Apr 16 '18 at 21:16

3 Answers3

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Well since $n > k \implies n \ne k$ then just say $n > k$. The $n \ne k$ will be implied and need not be stated at all.

fleablood
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  • So there is absolutely not a symbol? No formal or informal symbol that has ever been used to represent both $n \neq k$ and $n > k$? – supersmarty1234 Apr 16 '18 at 22:25
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    No, because every person who has ever walked the planet had realized that if $n>k$ means that $n\ne k$. – fleablood Apr 16 '18 at 22:31
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Since $n > k$ implies $n \not = k$, we find that "$n > k$ or $n \not = k$" is equivalent to simply $n \not = k$

In logic:

We have that $$n \not = k \Leftrightarrow n > k \lor n < k$$

Hence:

$$n > k \lor n \not = k \Leftrightarrow n > k \lor n > k \lor n < k\Leftrightarrow n > k \lor n < k \Leftrightarrow n \not = k$$

EDIT

I see that the exercise now has become how to write "$n > k$ or $n \not = k$". Well:

$$n > k \land n \not = k \Leftrightarrow n > k \land (n > k \lor n < k) \overset{Absorption}{\Leftrightarrow} n > k$$

Of course! If $n > k$ then $n \not = k$, and so $n \not = k$ is not adding anything to the statement $n >k$

Bram28
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As others have said, that's what $\;>\;$ is for. However, if you want to draw special attention to the fact that $\;\neq\;$ applies, then you can use $\;\gneqq \;$ or $\;\gvertneqq \;.$

From the mid 1970s through the early 1990s, I sometimes used (in notes, in homework assignments, etc.) handwritten forms of these symbols and their "less than" versions $\;\lneqq\;$ and $\;\lvertneqq \,,\;$ and they were also sometimes used on the blackboard by the mathematics teachers I had during this period.

Incidentally, even more commonly used back then was the corresponding proper subset relation $\subsetneqq \,,\;$ since the meaning of $\;\subset\;$ varies among authors, with some using this for "subset or equal" and others using this for "proper subset". By using $\;\subsetneqq\;$ for "proper subset" and using $\;\subseteqq \;$ or $\;\subseteq \;$ for "subset or equal", there is no chance of ambiguity.

For what it's worth, these symbols are available in the LaTeX-based software I use, Scientific Workplace. However, they do not appear to be available in MathType (which I sometimes have to use for work-related stuff), but maybe doing something like this will locate them. A quick google image search led me to the Mathematics Stack Exchange question What does this “double less than or equals to” sign mean?, where both $\;\gneqq \;$ and $\;\lneqq\;$ are included in a chart for AMS codes for various inequality symbols.

Dave L. Renfro
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