On a test I wrote an implication arrow "$\implies$" to show that I deduced one statement from the previous one, but I didn't get full score since it was more accurate to use an equivalence arrow "$\iff$".
For example:
$$ 2x = 4 \implies x = 2 $$
but it's also true the other way around:
$$ 2x = 4 \impliedby x = 2$$
so it is more correct to write equivalence arrow:
$$ 2x = 4 \iff x = 2$$
Given this i would assume that if $Q \implies P$ is true, then $Q \impliedby P$ is false.
Is this correct?
I don't want to check whether a statement only implies or is equivalent to another every time I do some operations to it.
So my second question is then: is there some other more loosely defined implication arrow that allows me to show that implication in one direction is true, without saying that implication the other direction is false? I also came across this picture, but i'm not entirely sure what the difference between those two definitions are.
Best Answer
As your own very example shows: just because the implication goes one way doesn't mean that it doesn't go the other way as well. In your case, it goes from left to right and from right to left, so we can write $P \Leftrightarrow Q$. But this does not mean that one of $P \Rightarrow Q$ or $Q \Rightarrow P$ is false. In fact, both would be true!
There is no commonly used symbol to say that you only have a one-way implication ... You'd have to explicitly say "$P \Rightarrow Q$ but not $Q \Rightarrow P$" ... Or $P \Rightarrow Q$ and $Q \not \Rightarrow P$