[avip]

The conclusion does not follow...

(the missing condition is that every hand is beatable)

[personjerry]

There are 5 possible hands

Each hand beats 2 distinct hands

With 3 distinct hands you can cover up to 6 distinct hands which is a superset of all possible hands

3 * 2 > 5

[avip]

And yet...

  A --> D | E
  B --> D | E
  C --> D | E
  D --> C | E
  E --> D | C

QED

[personjerry]

Um, this isn't a general math problem; We're not solving for all possible sets. We're solving this problem in particular:

  A --> B | C
  B --> C | D
  C --> D | E
  D --> E | A
  E --> A | B