Given the relation $r = \{(i, j) : |i - j| = 2\}$ on the set $\{1, 2, 3, 4, 5, 6\}$.

(a) Reflexive?

• Reflexive: $\forall i, (i, i) \in r$
• Check:
  • $|1-1| = 0$
  • $|2-2| = 0$
  • $|3-3| = 0$
  • $|4-4| = 0$
  • $|5-5| = 0$
  • $|6-6| = 0$
• Not Reflexive: No pairs $(i, i)$.

(b) Symmetric?

• Symmetric: $\forall (i, j) \in r, (j, i) \in r$
• Check pairs in $r$:
  • From $(1, 3) \to (3, 1)$ in $r$
  • From $(2, 4) \to (4, 2)$ in $r$
  • From $(3, 5) \to (5, 3)$ in $r$
• Symmetric: All pairs satisfy symmetry.

(c) Transitive?

• Transitive: $\forall (i, j) \in r, (j, k) \in r \rightarrow (i, k) \in r$
• Check pairs:
  • $(1, 3), (3, 5) \not\rightarrow (1, 5)$ in $r$
• Not Transitive: Fails at $(1, 3)$ and $(3, 5)$.

(d) Graph of $r$

1. Vertices: $1, 2, 3, 4, 5, 6$
2. Edges:
   • $(1, 3)$
   • $(2, 4)$
   • $(3, 1)$
   • $(3, 5)$
   • $(4, 2)$
   • $(5, 3)$

Graph Representation:

  1 -- 3 -- 5
  2 -- 4

Summary:

• (a) Not reflexive
• (b) Symmetric
• (c) Not transitive

Would you like further details or explanations? Here are some related questions to consider:

1. What is the definition of a reflexive relation?
2. Can you provide examples of symmetric relations?
3. How would transitivity apply to other relations?
4. How can you visualize relations differently?
5. What are the implications of symmetry in graph theory?

Tip: Always check each property individually and use examples to clarify understanding.