Difference between $\color{red}{\implies}$, $\color{red}{\Longleftarrow}$, and $\color{red}{\Longleftrightarrow}$ and when they use each?

Question

I often encounter the logic symbols $\implies$, $\Longleftarrow$, and $\Longleftrightarrow$ in mathematical text. Personally, I frequently use $\implies$ when doing mathematics, but I am curious in what situations each should be correctly used.

Although I frequently use $\implies$, I have realised that I often do it incorrectly; for instance, I am sometimes told that I should not have used an implication ($\implies$) symbol, because the equation is not necessarily directly implied (or something along those lines). In the context of mathematics, it seems that my use of these symbols is sloppy, which itself signals a lack of understanding.

What is the difference between $\implies$, $\Longleftarrow$, and $\Longleftrightarrow$ in mathematics, and when is it appropriate/correct to use each? Please speak about this in the context of elementary (high school and early university) mathematics so that it is easily understandable.

I would greatly appreciate it if people could please take the time clarify these concepts.

Answer 1

The use of $\implies$ is appropriate if what follows it is a consequence of what precedes it. For example, $$x=5 \implies x^2=25$$ is a proper use. However in most contexts $$x^2=25 \implies x=5$$ is an invalid use, as from $x^2=25$ you'd not be able to conclude that $x=5$, since also $(-5)^2=25$. Note however in a context where you've established that $x$ is a natural number, the above use is valid, as in that case indeed, from $x^2=25$ you can conclude that $x=5$ (since $-5$ is not a natural number).

$A\impliedby B$ is exactly the same as $B\implies A$, therefore e.g. $$x^2=25 \impliedby x=5$$ is a correct usage. You can use it if you for some reason want to tell the implication first. Which of the forms you use is a matter of style, as long as you make sure that the conclusion is on the pointy end of the arrow.

$A\iff B$ means that both $A\implies B$ and $B\implies A$ (resp. $A\impliedby B$). It is used if either side follows from the other. For example, $$x^2=25 \iff x=-5 \lor x=5$$

Answer 2

"$P\Longleftarrow Q$" is defined to mean "$Q\Longrightarrow P$", and "$P\iff Q$" is defined to mean "$P\Longrightarrow Q$ and $Q\Longrightarrow P$". So all of these symbols can be expressed in terms of the basic symbol "$\Longrightarrow$".

Its meaning is simple assertion of implication. "$P\Longrightarrow Q$" is just the statement that whenever $P$ is true, $Q$ is also necessarily true. It does not assert that either of these is true, just that the truth of one is a consequence of the truth of the other whenever that should occur.

Answer 3

It's conventional using $\Longleftrightarrow$ instead of iff (an abriviation of if and only if), only say that in statement we usualy used iff and when we write mathematical statement we can use $\Longleftrightarrow$ like

$n$ is even iff $n+1$ is odd.

$k|n\Longleftrightarrow\exists\ell\in\mathbb{Z}:n=k\ell$