Relations

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A relation R on set A = {1, 2, 3, 4} is defined as R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1)}. Which property does R satisfy?

RankGuru · Accepted Answer

1. **Reflexive Check**: Every element $a \in A$ must have $(a, a) \in R$.
   - $(1,1), (2,2), (3,3), (4,4)$ are all present. **Reflexive**.
2. **Symmetric Check**: If $(a, b) \in R$, then $(b, a)$ must be in $R$.
   - $(1,2)$ is in $R$, and its mirror $(2,1)$ is also in $R$. **Symmetric**.
3. **Transitive Check**: If $(a, b)$ and $(b, c) \in R$, then $(a, c)$ must be in $R$.
   - For $(1,2)$ and $(2,1)$, we need $(1,1)$, which is present.
   - For $(2,1)$ and $(1,2)$, we need $(2,2)$, which is present.
   - Other diagonal pairs don't generate new requirements. **Transitive**.
4. **Conclusion**: $R$ satisfies all three, making it an **equivalence relation**.

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