21.4.4. Part D: Irreflexive Relations

Try these first

1. 1.

   Give an example of an irreflexive relation on the set {x, y}.

   Hint

   Avoid including (x, x) or (y, y).

2. 2.

   What is the definition of an irreflexive relation?

   Hint

   Think about the pairs that are disallowed.

3. 3.

   An irreflexive relation cannot include which of the following pairs?

   • (a
   • a)
   • (a
   • b)
   • (b
   • a)

   Hint

   Focus on the definition of irreflexive relations.

4. 4.

   True or False: An irreflexive relation can be symmetric.

   • True
   • False

   Hint

   Think about simultaneous inclusion conditions.

5. 5.

   Given a set of n elements, derive the conditions under which an irreflexive and symmetric relation can exist.

   Hint

   Focus on distinguishing unique pairs against their reversals.

6. 6.

   Formulate a proof that if a relation is both irreflexive and anti-symmetric, it must only consist of pairs where the two elements are distinct.

   Hint

   Analyze how each condition limits the inclusion of pairs distinctly.