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From two propositions $P$ and $Q$, we can construct the implication $P \Rightarrow Q$. It is well-known that, the truth of $P\Rightarrow Q$ does not imply that the converse $Q \Rightarrow P$ is also true. When $P\Rightarrow Q$ and $Q \Rightarrow P$ are both true, we denote it by an equivalence $P \Leftrightarrow Q$.

Out of curiosity, is there a (relatively widespread) notation to mean that
"$P\Rightarrow Q \textrm{ and } \textrm{not}(Q\Rightarrow P)$" is true. And, if possible, please add a Latex command to represent this symbol.

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    I sometimes see things like this: P$\begin{smallmatrix}~~\not\!\!\!\!\impliedby\\\implies\end{smallmatrix}Q$, but you could just as easily write it as $\neg P \vee Q$ – 2017-02-20
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    Well, I don't know if this page contains your answer, but [this page](https://en.wikipedia.org/wiki/List_of_logic_symbols) contains a lot of logic symbols. Perhaps it could be helpful. – 2017-02-20
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    In words, you might say that $Q$ is necessary, but not sufficient for $P$. – 2017-02-20

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