(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:
