21.5.1. Part A: Symmetric, Anti-Symmetric and Reflexive Relations

Try these first

1. 1.

   Provide an example of a symmetric relation.

   Hint

   Think of relationships where mutual conditions apply.

2. 2.

   Define a reflexive relation.

   Hint

   What condition must always hold true in a reflexive relation?

3. 3.

   What defines a symmetric relation?

   • If (a
   • b) implies (b
   • a)
   • If (a
   • b) implies (a
   • c)
   • If no elements relate to themselves

   Hint

   Think about relations that require mutual connections.

4. 4.

   True or False: An anti-symmetric relation can have (a, b) and (b, a) both present if a does not equal b.

   • True
   • False

   Hint

   Remember the definition of anti-symmetry.

5. 5.

   Construct a relation on 4 elements that is symmetric yet not reflexive. Describe your relation.

   Hint

   What must you include while maintaining symmetry and excluding reflexivity?

6. 6.

   Prove that any reflexive relation on a set is not anti-symmetric if it includes all pairs.

   Hint

   Look closely at the definitions and what full inclusion means.