21.5.2. Part B: Symmetric, Anti-Symmetric and Irreflexive Relations

Try these first

1. 1.

   Define a symmetric relation in your own words.

   Hint

   Think about how relationships often work in pairs.

2. 2.

   What is an example of an irreflexive relation?

   Hint

   Consider numerical comparisons where one number can’t equal itself.

3. 3.

   What defines a symmetric relation?

   • If (a
   • b) implies (b
   • a)
   • If (a
   • b) implies b > a
   • If (a
   • b) implies (a
   • a)

   Hint

   What would happen if you reversed the pairs in a relationship?

4. 4.

   True or False: An anti-symmetric relation can have (a, a) and (b, b) without issue.

   • True
   • False

   Hint

   Consider the definition of anti-symmetry carefully.

5. 5.

   Given a set A = {a, b, c}, construct all possible relations and categorize them by symmetry, anti-symmetry, and irreflexivity.

   Hint

   Use the definition of each type to guide your categorization.

6. 6.

   Prove that a symmetric relation on a set can never be irreflexive.

   Hint

   Start with the definitions of symmetric and irreflexive relations.