17.6. Summary of Binary Relations

Try these first

1. 1.

   Is the relation R = {(1, 2), (2, 3)} irreflexive?

   Hint

   Check if (a, a) is in R.

2. 2.

   What defines a symmetric relation?

   Hint

   Think of reciprocal relationships.

3. 3.

   What defines an irreflexive relation?

   • A must relate to itself
   • A cannot relate to itself
   • A can relate to any element

   Hint

   Think about whether (a, a) is present.

4. 4.

   True or False: A transitive relation means if a relates to b and b to c, then a must relate to c.

   • True
   • False

   Hint

   Visualize the relationships in a graph.

5. 5.

   Can a relation be both irreflexive and reflexive for a non-empty set? Justify your answer.

   Hint

   Reflect on the definitions of both properties.

6. 6.

   Provide a detailed example of a relation that satisfies all three properties: symmetric, antisymmetric, and transitive. Is such a relation possible?

   Hint

   Consider the characteristics of an empty set.