The truth table of $p\implies q$ is as follows:
$$\begin{array}{c|c|c} \boldsymbol{p} & \boldsymbol{q} & \boldsymbol{p\implies q} \\ \hline \color{green}{\text{T}} & \color{green}{\text{T}} & \color{green}{\text{T}} \\ \hline \color{green}{\text{T}} & \color{red}{\text{F}} & \color{red}{\text{F}} \\ \hline \color{red}{\text{F}} & \color{green}{\text{T}} & \color{green}{\text{T}} \\ \hline \color{red}{\text{F}} & \color{red}{\text{F}} & \color{green}{\text{T}} \end{array}$$
I know it's just a definition, but I'd like to know the motivation behind it. Here's what I think:
Strictly speaking, the meaning of the proposition $p\implies q$ can be interpreted as if $\boldsymbol p$ is true, then $\boldsymbol q$ must be also true. This means that there's no case in which $\boldsymbol p$ is true and $\boldsymbol q$ is false. So if $p$ is true and $q$ is false, then $p\implies q$ becomes false and in other cases, it is true.
Am I right?