17.1.3. Reflexive and Irreflexive Relations

Try these first

1. 1.

   Define a reflexive relation on the set {a, b}.

   Hint

   Remember that every element must relate to itself.

2. 2.

   What is an example of an irreflexive relation on the set {x, y}?

   Hint

   Ensure none of the pairs (x, x) or (y, y) are included.

3. 3.

   What defines a reflexive relation?

   • All elements are related to one another
   • Every element is related to itself
   • No elements are related

   Hint

   Think about how elements relate to themselves.

4. 4.

   True or False: An empty set can be both reflexive and irreflexive.

   • True
   • False

   Hint

   Consider how there are no elements to contradict the definitions.

5. 5.

   Given set A = {1, 2, 3}, define a reflexive relation, then modify it to make it irreflexive without removing elements.

   Hint

   Reflexivity requires self-relations, irreflexivity denies them.

6. 6.

   Is it possible to have a single relation that is both reflexive and irreflexive? Explain your reasoning.

   Hint

   Consider what happens when you have no items at all.