Upon reading the part 1.4.1 Logical Deductions of the book Mathematics for Computer Science, I found the inference rules vaguely understood because only their notations are shown without concrete examples and I'm not sure if the examples I come up with by myself are legit or good enough. What are some typical examples for the following rules?
Rule 1.
Modus Ponens: $\frac{P,\space\space P\space IMPLIES\space\space Q} Q$
Rule 2.
$\frac{P\space\space IMPLIES\space\space Q,\space\space Q\space\space IMPLIES\space\space R} {P\space\space IMPLIES\space\space R}$
Rule 3.
$\frac{NOT(P)\space\space IMPLIES\space\space NOT(Q)} {Q\space\space IMPLIES\space\space P}$
Non-rule.
$\frac{NOT(P)\space\space IMPLIES\space\space NOT(Q)} {P\space\space IMPLIES\space\space Q}$