17.5.2. Examples of Transitive Relations

Try these first

1. 1.

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

   Hint

   Recall that an irreflexive relation cannot contain (a, a).

2. 2.

   Is R = {(1, 2), (2, 1)} symmetric?

   Hint

   Check if both directional relations are included.

3. 3.

   What is an irreflexive relation?

   • A relation where some elements relate to themselves
   • A relation where no elements relate to themselves
   • A relation with all elements relating to each other

   Hint

   Recall the definition of irreflexive.

4. 4.

   True or False: Every symmetric relation is also antisymmetric.

   • True
   • False

   Hint

   Think about how symmetry contradicts antisymmetry.

5. 5.

   Given a relation R = {(1, 2), (2, 1), (2, 3)}, determine if R is symmetric, antisymmetric, or both.

   Hint

   Examine the conditions set by the definitions.

6. 6.

   If a relation defined on a set A is both symmetric and antisymmetric, under what conditions does this occur?

   Hint

   Think about how each definition constrains relationships.