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Exercise 4 - Chapter 1

edited May 2018 in Exercises

Now that we're about to dip our toe into the sea of logic, it's good to do an exercise involving the Booleans:

$$ \mathbf{Bool} = \{ \textrm{true}, \textrm{false} \} . $$ This set becomes a poset where \(\textrm{false}\leq\textrm{false}\), \(\textrm{false}\leq\textrm{true}\), and \(\textrm{true}\leq\textrm{true}\), but \(\textrm{true}\not\leq\textrm{false}\). In other words \(A\leq B\) in the poset if \(A\) implies \(B\), often denoted \(A\implies B\).

In any poset \(A \vee B\) stands for the join of \(A\) and \(B\): the least element of the poset that is greater than both \(A\) and \(B\). The join may not exist, but it is unique.

In the poset \(\mathbb{B}\), what is

  • \(\textrm{true} \vee \textrm{false}\)?

  • \(\textrm{false} \vee \textrm{true}\)?

  • \(\textrm{true} \vee \textrm{true}\)?

  • \(\textrm{false} \vee \textrm{false}\)?

Comments

  • 1.
    edited April 2018

    This does not appear to be formatting correctly... try: \text{true} and \text{false}?

    Comment Source:This does not appear to be formatting correctly... try: <code>\text{true}</code> and <code>\text{false}</code>?
  • 2.
    edited April 2018

    Done.

    Comment Source:Done.
  • 3.
    edited April 2018

    We've been working out lots of "meets", or "least upper bounds", so it's good to emphasize that they don't always exist!

    Puzzle 26. What is the simplest poset that contains two elements \(x\) and \(y\) that have no least upper bound?

    Puzzle 27. What is the simplest poset that contains a subset that has no least upper bound?

    Comment Source:We've been working out lots of "meets", or "least upper bounds", so it's good to emphasize that they don't always exist! **Puzzle 26.** What is the simplest poset that contains two elements \\(x\\) and \\(y\\) that have no least upper bound? **Puzzle 27.** What is the simplest poset that contains a subset that has no least upper bound?
  • 4.
    edited April 2018

    Hi John Baez, where above you wrote \( A \vee B \) means the meet, did you mean, means the join?

    Comment Source:Hi [John Baez](https://forum.azimuthproject.org/profile/17/John%20Baez), where above you wrote \\( A \vee B \\) means the meet, did you mean, means the join?
  • 5.

    Yes, I meant "join", and I will fix this. I often mix up the words "meet" and "join" because in ordinary language the mean similar things. I find it easier to remember "greatest lower bound" and "least upper bound", or "inf" and "sup".

    Comment Source:Yes, I meant "join", and I will fix this. I often mix up the words "meet" and "join" because in ordinary language the mean similar things. I find it easier to remember "greatest lower bound" and "least upper bound", or "inf" and "sup".
  • 6.
    edited April 2018

    Re. #5 I always remember the Open Systems Interconnect layers as PuDiN TSPA.

    Q. Are these mnemonics correct even if not useful?

    • Meet up (^) and greatest intersect (MUAGI).
    • Join down (v) or least disjunct (JDOLD).
    Comment Source:Re. #5 I always remember the Open Systems Interconnect layers as PuDiN TSPA. Q. Are these mnemonics correct even if not useful? * Meet up (^) and greatest intersect (MUAGI). * Join down (v) or least disjunct (JDOLD).
  • 7.
    edited April 2018

    Jim: the great antimnemonic is this picture:

    image

    The meet \(\wedge\) is at the bottom and the join \(\vee\) is at the top, exactly opposite from how their symbols look!

    Comment Source:Jim: the great _antimnemonic_ is this picture: <center><img src = "https://upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Join_and_meet.svg/500px-Join_and_meet.svg.png"></center> The meet \\(\wedge\\) is at the bottom and the join \\(\vee\\) is at the top, exactly opposite from how their symbols look!
  • 8.

    Lol. :)

    Comment Source:Lol. :)
  • 9.
    edited April 2018

    For me it's mnemonic*; I think of the Meet (Join) symbol as pointing up (down) to the elements in the subset (and the up-arrow occurs twice in M, while the down-arrow is vaguely J-shaped).

    *I hope this doesn't cancel w/ the anti-mnemonic; you should probably try to forget at least one of these:-)

    Comment Source:For me it's mnemonic*; I think of the Meet (Join) symbol as pointing up (down) to the elements in the subset (and the up-arrow occurs twice in M, while the down-arrow is vaguely J-shaped). *I hope this doesn't cancel w/ the anti-mnemonic; you should probably try to forget at least one of these:-)
  • 10.

    I think of it as diving crocodile and soaring crocodile. :)

    Comment Source:I think of it as diving crocodile and soaring crocodile. :)
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