What does $\rightarrow$ mean in $p \rightarrow q$

Solution 1:

The $\rightarrow$ symbol is a connective. It's a symbol which connects two propositions in the context of propositional logic (and its extensions, first-order logic, and so on).

The truth table of $\rightarrow$ is defined to be that $p\rightarrow q$ is false if and only if $p$ is true and $q$ is false.

Indeed this is the same meaning of $\implies$, but the difference is that $p\implies q$ is a statement about propositions, whereas $p\rightarrow q$ is a proposition. In some contexts, though, people don't make this distinction between material implication (the connective) and logical implication (the $\implies$ arrow). But they are not the same thing in every context of propositional logic.

Solution 2:

It is a material conditional, or otherwise known as $p$ implies $q$, or if $p$, then $q$

The truth table for that is as follows

p  q  p implies q
T  T  T 
T  F  F
F  T  T
F  F  T

$\rightarrow$ can also be written as $\implies$.

In computer science, $p \implies q$ can be rewritten as (not p) or q, or !p||q

Solution 3:

Given $p$, then we have $q$.

or $p$ implies $q$.

The two arrows mean the same thing.