Relations
Antisymmetric Relations
Premium Question
Grade 11

Question:

A relation $R$ on $A$ is both symmetric and antisymmetric. Then $R$ is a subset of:
(1) $A \times A$
(2) $\{(a, a) : a \in A\}$
(3) $A$
(4) $\emptyset$

Step-by-Step Solution

Key Concept: Symmetric: $(a, b) \in R \Rightarrow (b, a) \in R$. Antisymmetric: $(a, b) \in R$ and $(b, a) \in R \Rightarrow a = b$. Combining: every element of $R$ must be of the form $(a, a)$. So $R \subseteq \{(a, a) : a \in A\}$ (the identity/diagonal relation).
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Correct Answer: (2)

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