Implication and equivalence arrows, when to 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 $\mathrm{even}(x)\Leftrightarrow\lnot\;\mathrm{odd}(x).$

Suppose you have two propositions $P$ and $Q$.

• $P\Rightarrow Q$ means that $P$ implies $Q$ (or if $P$, then $Q$).
• $P\Leftrightarrow Q$ means that $P$ implies $Q$ and $Q$ implies $P$ (or $P$ if and only if $Q$).

Let me add a few simple examples.

1. Since $\frac{2p}{2q}=\frac{p}{q}$ ($q\ne 0$) we can write $$\frac{2p}{2q}=5\Leftrightarrow \frac{p}{q}=5.\qquad (1)$$

   Its meaning is that $\frac{2p}{2q}=5$ is true if and only if $\frac{p}{q}=5$.

2. a) If you have $y=x$, then you can say that $y^2=x^2$ and write $$y=x\Rightarrow y^2=x^2.\qquad (2)$$

   Its meaning is that if $y=x$ is true so is $y^2=x^2.$

   b) If you have $y=-x$, then you can say that $y^2=x^2$ and write

   $$y=-x\Rightarrow y^2=x^2.\qquad (3)$$

   Its meaning is that if $y=-x$ is true so is $y^2=x^2.$

   c) If you know that $y^2=x^2$ you can guarantee that $y=x$ or $y=-x$ and write

   $$y^2=x^2\Rightarrow y=x\quad \text{or}\quad y=-x.\qquad (4)$$

   Its meanig is that if $y^2=x^2$, then $y=x$ or $y=-x$.

   d) Combining $(2),(3)$ and $(4)$ you have the following equivalence

   $$y^2=x^2\Leftrightarrow y=x\quad \text{or}\quad y=-x.\qquad (5)$$

   Its meanig is that $y^2=x^2$ if and only if $y=x\quad \text{or}\quad y=-x$.