### Notes

(This is where I put fragments of ideas for future tutorials)

 Name Statement Implication if p, then q Converse if q, then p Inverse if not p, then not q Contrapositive if not q, then not p

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:

 p q p => q q => p !p => !q !q => !p 1 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 1