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Among reflexive, symmetric, antisymmetric, and transitive, which of those properties are true of the above relation?

  1. It is both reflexive and symmetric
  2. It is only reflexive
  3. It is only antisymmetric
  4. It is both reflexive and transitive
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The relation $R$ is define over set $S=\left \{0,1,2,3 \right\}$ is as follows:

$R=\left\{ (0,0) (1,1) (2,2) (3,3 ) (0,1) (1,0) (1,2) (2,1) (2,3) (3,2)\right\}$

  • $R$ is reflexive relation as all digonal elements is present; selfloop at each node.
  • $R$ is symmetric relation as bidirectional edges present between every node; $(0,1)\in R\rightarrow (1,0)\in R$. we can check other possibility also.
  • $R$ is not antisymmetric as $(0,1) ,(1,0)$ both pairs is present in the relation.
  • $R$ is not transitive relation because $(0,1)\in R,(1,2)\in R\rightarrow(0,2)\notin R$

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