17.2.3. Reflexive vs. Symmetric Relations

Try these first

1. 1.

   What is an irreflexive relation?

   Hint

   Think about what happens with self-loops.

2. 2.

   Can a relation be both reflexive and irreflexive? Under what condition?

   Hint

   Consider the implications of having no elements.

3. 3.

   What is a symmetric relation?

   • A. If (a
   • b) exists
   • then (b
   • a) must also exist.
   • B. A relation with no self-loops.
   • C. A relation that is always reflexive.

   Hint

   Think about pairs relating to each other.

4. 4.

   True or False: An empty relation can be considered symmetric.

   • True
   • False

   Hint

   Remember the conditions for relations in the absence of elements.

5. 5.

   Given a set A = {1, 2, 3, 4} and a relation defined as R = {(1, 2), (2, 1), (3, 4)}, determine if R is symmetric, asymmetric, or neither.

   Hint

   Identify the pairs and their relationships.

6. 6.

   Prove or disprove: A relation can be both asymmetric and antisymmetric.

   Hint

   Consider the definitions closely.