Relations
Equivalence Relations
AIEEE 2005
Grade 11
Question:
Let $R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\}$ on $A = \{3, 6, 9, 12\}$. $R$ is:
(1) reflexive and transitive but not symmetric
(2) reflexive and symmetric but not transitive
(3) symmetric and transitive but not reflexive
(4) an equivalence relation
Step-by-Step Solution
Key Concept: All $(a, a)$ are present $\Rightarrow$ reflexive. $(6, 12) \in R$ but $(12, 6) \notin R \Rightarrow$ not symmetric. Check transitivity by exhaustion: all compositions $(a, b), (b, c) \in R$ yield $(a, c) \in R$ (e.g. $(3, 6), (6, 12) \to (3, 12)\checkmark$). So reflexive and transitive but not symmetric.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)