(This is where I put fragments of ideas for future tutorials)
Let's consider an example. Take two statements:
p = Manchester United won every game of the season
q = Manchester United won the league
p implies q -- if MU won every game of the season, then they will definitely have won the league. Is the converse true - i.e. does q imply p? In other words, if MU won the league, does it imply that they won every game? Is this true -- No! What about the inverse statement? In other words, if MU have not won every game of the season, then they have not won the league? Is this true -- not necessarily! Finally, let's consider the contrapositive: If MU haven't won the league, then they haven't won every game of the season. Is this statement true -- Yes!
How do we establish that the contrapositive is equivalent to the original statement in all cases? One way is using a truth table: