Relations
Types of Relations
JEE Main 2018
Grade 11
Question:
Let $A = \{a, b, c\}$ with $R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}$ and $R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}$. Then:
(1) $R_1$ is not symmetric, $R_2$ is not transitive
(2) $R_1$ symmetric, $R_2$ not transitive
(3) Both $R_1$ and $R_2$ are symmetric
(4) $R_1$ not reflexive, $R_2$ reflexive
Step-by-Step Solution
Key Concept: $R_1$: $(b, c) \in R_1$ but $(c, b) \notin R_1 \Rightarrow$ not symmetric. $R_2$: $(b, a), (a, b) \in R_2$ but $(b, b)\checkmark$; however $(b, a), (a, c) \in R_2 \Rightarrow$ need $(b, c)$, which is absent $\Rightarrow$ not transitive. Answer: (1).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)