Implication and equivalence arrows, when to use them? In my course book we have something called implication arrows $\Rightarrow$ and equivalence arrows $\Leftrightarrow$ and I have never managed to understand them. 

When do I know which to use and how do I know that I'm correct when I use them?
$A \Rightarrow B$ has the meaning $B$ follows from $A,$ but this does not necessarily hold the other way around. For instance, from $x = 13$, it follows that $x$ is prime. But you can't argue the other way around. So if $\mathbb P$ is the set of prime numbers, you can write $x = 13 \Rightarrow x\in\mathbb P$, but not $x\in\mathbb P\Rightarrow x=13.$

$A\Leftrightarrow B$ has the meaning that $A$ and $B$ are the same statement, just transformed.  If $A\Leftrightarrow B$, then both $A\Rightarrow B$ and $B\Rightarrow A.$  For instance, consider an integer $x$.  You can say that if $x$ is even, then $x$ is not odd.  This certainly also holds the other way around, so you can write $\mathrm{even}(x)\Leftrightarrow\lnot\;\mathrm{odd}(x).$
I find the implication of logical implication really interesting. Starting off with the truth tables for the two, note only one subtle difference:

+---+---+-------+
| P | Q | P ⇒ Q |    
+---+---+-------+
| T | T |     T |
| T | F |     F |
| F | T |     T |
| F | F |     T |
+---+---+-------+

+---+---+-------+
| P | Q | P ⇔ Q |    
+---+---+-------+
| T | T |     T |
| T | F |     F |
| F | T |     F |
| F | F |     T |
+---+---+-------+



There is a common misconception that P ⇒ Q means that if P is true, Q is true.  What the truth table shows is that P ⇒ Q is always true when P is false, which I think has profoundly interesting implications. 

In some ways the truth table is simply a formal version of the old computer cliche "garbage in, garbage out": P actually tells us nothing about Q.

Q, on the other hand, if we know it is false, tells us if P is true or not provided we know if P ⇒ Q is true or false. But if Q is true, it gives us no clue as to whether P is true or false.

It is a lesson that we can only advance knowledge by working back from known wrong conclusions. There is no way to link known correct conclusions to what we think causes them with logical certainty.

This may seem obvious to lawyers and detectives, but it is very counterintuitive to us computer programmers who think of P as a pattern to be matched by an if statement, and Q as the resulting action.

My view is logical implication creeps up in other fields such as the old statistics cliche "correlation does not imply causation" and Karl Popper's scientific method of falsifiability. But I've never really seen this articulated anywhere.

Through deduction we can whittle down the choice of possible causes, Ps, for a given Q until the answer seems obvious, but even then it is never 100% certain. 

Logical implication encourages an open mind which remembers “there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy”. Many zealots out there could have been cured if our basic education system taught logical implication at an early age.