Notation for “A does not imply B”

I'm looking for a way to express A being true does not imply B. I know that A implies B can be written as $A \rightarrow B$, but what about A does not imply B? $A \not\rightarrow B$?

• Simply write $A\land\neg B$, though that is probably not what you are after. I think you want something like "$p$ is prime does not imply in general that $p$ is odd", right? – Hagen von Eitzen Nov 17 '12 at 17:38
• @BabakSorouh That tells me how to write the symbol in LaTeX, but not what the symbol means (except possibly by its name). – gerrit Nov 17 '12 at 17:56
• @HagenvonEitzen Yes, it's the latter I'm after. – gerrit Nov 17 '12 at 17:57
• I just came here to learn to write $can \nRightarrow should$ – Ben May 2 '17 at 19:50

If you use a generic $\to$ or $\Rightarrow$ for imply then slash through for the not-imply.

But you can perhaps do better. If you actually mean syntactic entailment (so non-implication is a matter of there being no proof from $A$ to $B$ in the relevant proof system) then $A \nvdash B$ is available and absolutely standard.

If you actually mean semantic entailment (so non-implication is a matter of there being a valuation which makes $A$ true without making $B$ true) then $A \nvDash B$ is available and quite standard.

• Is there a way to formulate $A \nvdash B$ with just and's, or's, and not's? – Matt Sep 14 '16 at 19:19

Usually, we use double arrows for implications: $A\Rightarrow B$. You can use a crossed out double arrow for does not imply: $A\nRightarrow B$. In LaTeX, these are "\Rightarrow" and "\nRightarrow", respectively.

• I would usually see single arrow $A \rightarrow B$ and negation $A \nrightarrow B$. – Andrew Wonnacott May 23 at 21:04