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Exercise 108 - Chapter 1

edited June 2018 in Exercises

Draw the Hasse diagram for the preorder \( \mathbf{Rel}( \{ 1, 2 \} ) \) of all binary relations on the set \( \{ 1, 2 \} \).

Comments

  • 1.
    edited April 2018

    Is this what is intended?

    Hasse

    Comment Source:Is this what is intended? ![Hasse](https://docs.google.com/drawings/d/e/2PACX-1vTCgcacsG5ZRLBZqU1ZDLiM4Swx0ifCHDRRLS_GR8GN-jplFAjS_sAcNuyMrytKFgwqQplizbFjqHew/pub?w=746&h=375)
  • 2.

    Fredrick: This is my understanding of the question and how I have the answer written in my notebook. It seems like it would be the same Hasse diagram as for the powerset of a set of 4 elements. Since with \(S=\{1,2\}\), \(S\times S=\{(1,1),(1,2),(2,1),(2,2)\}\).

    Comment Source:Fredrick: This is my understanding of the question and how I have the answer written in my notebook. It seems like it would be the same Hasse diagram as for the powerset of a set of 4 elements. Since with \\(S=\\{1,2\\}\\), \\(S\times S=\\{(1,1),(1,2),(2,1),(2,2)\\}\\).
  • 3.

    @JaredSummers I guess the point was noticing that \( S \times S \) forms a set of 4 elements. I suppose it makes sense in the context of the subsequent exercise.

    Comment Source:@JaredSummers I guess the point was noticing that \\( S \times S \\) forms a set of 4 elements. I suppose it makes sense in the context of the subsequent exercise.
  • 4.

    Great picture, Fredrick! Yes, a binary relation on \(S\) is the same as a subset of \(S \times S\), so the poset of binary relations on \(S \) is just our friend the power set \( P(S \times S\). So, it looks like an \(n^2\)-dimensional cube if \(S\) has \(n\) elements. You're taking \(n = 2\) so your picture looks a lot like this:

    image
    Comment Source:Great picture, Fredrick! Yes, a binary relation on \\(S\\) is the same as a subset of \\(S \times S\\), so the poset of binary relations on \\(S \\) is just our friend the power set \\( P(S \times S\\). So, it looks like an \\(n^2\\)-dimensional cube if \\(S\\) has \\(n\\) elements. You're taking \\(n = 2\\) so your picture looks a lot like this: <center><img src = "http://math.ucr.edu/home/baez/mathematical/7_sketches/P4_hasse_diagram.png"></center>
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