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Amongst the properties $\left\{\text{reflexivity, symmetry, anti-symmetry, transitivity}\right\}$ the relation $R=\{(x, y) \in N^2|x \neq y\}$ satisfies _________
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  • It is not reflexive as $xRx$ is not possible.
  • It is symmetric as if $xRy$ then $yRx$.
  • It not antisymmetric as $xRy$ and $yRx$ are possible and we can have $x \neq y$.
  • It is not transitive as if $xRy$ and $yRz$ then $xRz$ need not be true. This is violated when $z = x$.

So, symmetry is the answer.

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Is this a valid answer

 

as x!=y Reflexive never holds true

For symmetry R={(1,2),(2,1)...} is possible

Having Symmetry invalidates anti-symmetry

Transitive is not possible as (1,2),(2,1) should imply (1,1) which we cannot get as it is not reflexive hence transitivity fails
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$\therefore$ The given relation  satifies only $\large symmetry$

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