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There may be many ways to do this. Are these three ways to represent reflexive relation using set builder notation? If not, the how to represent

$$\{a | a \in A, (a, a) \in R\} -(1)$$

$$Or$$

$$\{\forall a | a \in A \rightarrow (a, a) \in R\}- (2)$$

$$Or$$

$$ \forall a \{ a \in A \rightarrow (a, a) \in R\}- (3)$$

Ubi.B
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1 Answers1

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Assuming you are asking for "the set of all reflexive relations on $A$," then you'd want $$\{R \in \mathcal{P}(A\times A) \mid \forall a(a \in A \to (a,a)\in R)\}$$

Of your choices, only (1) properly represents a set. If we're given a set $A$ and a relation $R$ on $A$ then we have $A = (1)$ if and only if $R$ is reflexive, but that's the most you could say.

Brian Moehring
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  • Can you help me know that why we don't write relation definition in set builder notation? Why we write is as just in predicate form? Even though they represent a set (of a relation). – Ubi.B Jul 31 '20 at 03:35