So I asked this question before without getting a solid answer. I went and studied a bit more about binary relations and reflexive relations. I understand the theory, but am unsure about whether my application of the the theory is correct in my answers.

For the Question below:

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 less than one week apart. Determine:

(i) Whether or not the relation $R$ is reflexive;

I have stated that, $xRy$ is NOT reflexive, as for the relation $xRy$, ∀ $x∈A$ is not equal ∀ $y∈A$

is this right? if not could someone tell me why not and possibly show me how to get the right answer for this particular question.

  • $\begingroup$ Archimedes and Archimedes were born less than a week apart. So were John Lennon and John Lennon. And so on. $\endgroup$ – André Nicolas Nov 17 '15 at 16:42
  • $\begingroup$ oh ok, but thats taking the relation xRx, I assumed the question asked for xRy, so Archimedes wasn't born a week apart from John Lennon (x and y being different people) $\endgroup$ – roughosing Nov 17 '15 at 16:52
  • $\begingroup$ as you can see I'm really not fully comprehending this question.... $\endgroup$ – roughosing Nov 17 '15 at 16:52
  • $\begingroup$ You are maybe confusing reflexive ($xRx$) with symmetric (if $xRy$ then $yRx$). The relation born less than a week apart is also symmetric, but that is not what is being asked. $\endgroup$ – André Nicolas Nov 17 '15 at 16:58
  • $\begingroup$ so reflexivity only ever compares a set to itself? $\endgroup$ – roughosing Nov 17 '15 at 17:01

It is reflexive!

$xRx$ is: are you born less than one week apart... from yourself? But of course.

  • $\begingroup$ ok so it might be I'm misunderstanding of the phrasing of the question, why did you take the relation xRx, when the question asks for the relation xRy? $\endgroup$ – roughosing Nov 17 '15 at 16:48
  • $\begingroup$ @roughosing because reflexivity only asks about $xRx$. $R$ is the relation. $\endgroup$ – mrp Nov 17 '15 at 16:54
  • $\begingroup$ oh, so would i have to state that $xRx$ is reflexive, and $yRy$ is reflxive, or just state the one? $\endgroup$ – roughosing Nov 17 '15 at 16:56
  • $\begingroup$ It's the same statement. When you write $x$, you mean $\forall x$ $\endgroup$ – AnalysisStudent0414 Nov 17 '15 at 17:04

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