21.4.6. Part F: Neither Reflexive nor Irreflexive Relations

Try these first

1. 1.

   Define a reflexive relation.

   Hint

   Think about pairs (x, x) for elements in the set.

2. 2.

   What makes a relation irreflexive?

   Hint

   Consider the opposite of reflexive.

3. 3.

   What must a reflexive relation always include?

   • (x
   • y)
   • (a
   • b)
   • (a
   • a)

   Hint

   Link to the definition of reflexivity.

4. 4.

   Is it possible for a relation to be both reflexive and irreflexive?

   • True
   • False

   Hint

   Consider the implications of both terms.

5. 5.

   Prove that a set relation cannot be both reflexive and irreflexive by using a set S of size n.

   Hint

   Focus on defining elemental constraints.

6. 6.

   Given a relation is defined such that for every set element, at least one diagonal element exists while excluding some, how would you categorize that set?

   Hint

   Consider pair distributions.