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Let A be the set of all people who have ever lived. For x, y ∈ A, xRy if and only if x and y were born at least 30 days apart

I want to determine whether the relation xRy is reflexive, transitive, antisymmetric, and/or symmetric so I can go on to determine whether it is a partial order an equivalence relation, or both by means of simple counterexamples or short proofs.

  • So far the conclusion I have come to is that the relation xRx is not reflexive as x can not have been born 30 days apart from x

  • I also believe the relation xRy is symmetric as if x was born 30 days apart from y(xRy) then it follows that y was born 30 days apart from x (yRx)

I understand what it means for a relation to be an equivalence relation (symmetric, transitive, and reflexive) and what it means for a relation to be a partial order (antisymmetric, transitive, and reflexive) but I am unsure how to show these properties in this question.

I am unsure if the deductions I have made are correct and I am also unsure how to show if the relation is transitive and/or antisymmetric. Any help would be highly appreciated!

Renee Ofadu
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1 Answers1

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Your deductions about the reflexive and symmetric parts are correct. Note that the relation is not transitive. Assume $x$ and $y$ in $1000-1100$, and $z$ lived in $1130-1230$. Then $xRz$ and $zRy$, but $x$ is not related to $y$!

Note also that this relation is not antisymmetric. With the same example, $xRz$ and $zRx$, but $x\neq z$.

Jotabeta
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  • thank you for your help, I am trying to understand your transitive explanation, would it be valid to say for example x was born on 3rd Jan and y was born 5th March therefore xRy as they were born at least 30 days apart, then z was born 3rd Jan meaning yRz as they were born 30 days apart but xRz does not hold as they were not born 30 days apart hence this relation is not reflexive – Renee Ofadu Oct 28 '22 at 10:32
  • Yes, that is correct! – Jotabeta Oct 28 '22 at 10:37
  • thank you for your help, have a great day! – Renee Ofadu Oct 28 '22 at 10:41