Difference between $\Rightarrow$ and $\implies$ I've used both $\Rightarrow$ and $\implies$ interchangeably throughout my mathematics in school, and I want to know which is proper. When should I use $\Rightarrow$ over the implies arrow? Does it mean leads to? Say I have $x+3=4$. Would I say:
$x+3=4\Rightarrow x=1$
or
$x+3=4\implies x=1$
Since the equation doesn't really 'imply', it leads to the solution that $x=1$. I haven't been able to find anything on this online, so I would hope it could be cleared up here :)
 A: 

$x+3=4\implies x=1$
Since the equation doesn't really 'imply', it leads to the solution that $x=1$.


Take any $x.$ Satisfying $$x+3=4$$ implies that it also satisfies $$x=1.$$
That is, the solution set of the first equation is a (possibly proper) subset of the solution set of the second equation.
(So, $x^2=25\kern.6em\not\kern-.6em\implies x=5.)$

I've used both $\Rightarrow$ and $\implies$ interchangeably throughout my mathematics in school, and I want to know which is proper.

Did you mean to contrast $\Rightarrow$ and $\rightarrow$ instead?
$\Rightarrow$ and $\implies$ mean exactly the same: they are just different “handwriting”.
A: There are 3 different levels to consider:
1. Mathematical semantics: Both arrows mean "implies" or "from $A$ follows $B$".  If you'd use the symbols with paper+pencil or blackboard+chalk, they are merely indistinguishable, because noone would start measuring their length/height ratio.
2. Typography: Here is the main difference. The short $\Rightarrow$ fits better with inline text, wheras the longer $\implies$ goes better with separated formula and additional spacings like $$\text{it's raining} \quad\implies\quad \text{the streets are getting wet}$$
There are actually 3 variants of this arrow:

*

*$x\Rightarrow y\qquad$ x\Rightarrow y

*$x\Longrightarrow y\qquad$ y\Longrightarrow y

*$x\implies y\qquad$ x\implies y
As you can see, \Longrightarrow has less spacing around it than \implies.
3. Semantics of the LaTeX source: Glyphs named \implies or \iff can add semantics to the source code.  However, source semantics is far from being perfect, so that this point is only mentioned for completeness.
