2.6.1. Part (a): Symmetric and Transitive Relations

Try these first

1. 1.

   Define a symmetric relation with an example.

   Hint

   Think about how friendships work!

2. 2.

   What does it mean for a relation to be transitive?

   Hint

   Consider a chain of relationships!

3. 3.

   What is a symmetric relation?

   • If aRb then bRa
   • If aRb then aRc
   • No specific property

   Hint

   Recall definitions of relations.

4. 4.

   True or False: All transitive relations are also symmetric.

   • True
   • False

   Hint

   Consider the definitions carefully.

5. 5.

   Prove whether the relation defined by R = {(1, 2), (2, 3), (3, 1)} is transitive.

   Hint

   Check the connections between the elements carefully.

6. 6.

   Construct a relation R on a set A = {a, b, c, d} that is symmetric and give a counterexample to show why R can be transitive but not reflexive.

   Hint

   Always ensure to list all pairs thoroughly.