Sorry, I didn't check on this thread for a while! I and the people in my group are doing a mix of both; we are both working on real-world applications, which mostly means implementing and testing a bunch of techniques and ideas that are already out …
Puzzle 22. I think that \(X\) is "not \(Y\)" if \(X \wedge Y = \bigvee \emptyset\) (they have no elements in common) and \(X \vee Y = \bigwedge \emptyset\) (their union is the entire set).
Instead of referring explicitly to other elements of the p…
The first thing I notice is that \(\chi(m,n)\) can be replaced by a function of a single variable \(\psi(m-n)\). This also means that flipping the arguments to \(\chi\) just flips the sign of the single argument to \(\psi\). Is there a reason you ar…
It's totally necessary to specify the preorder set in which you're taking the meet, and not just when you're taking the meet of the empty set! The definitions of meet and join (definition 1.60 in the book) use the preorder set in the definitions. Fo…
I took 6.034 as an undergraduate long before the OCW version. :) I recognize around the first quarter of the current syllabus. There certainly may be other things on there that I have learned and forgotten, though.
Nice to see you here. I took Russ Tedrake's Underactuated Robotics online course (6.832x) a couple years back and really enjoyed it. I envied you guys who got to actually work with some hardware. (I also took Winston's AI course at MIT a looong time…
Just a note to encourage people to do all the exercises. They are "simple" enough that if you can't do them, you probably don't totally understand the material, so they're a useful check. They also act as nice speed bumps to slow you down a bit, for…
Aqilah: I don't think I understand your first question; can you rephrase? The empty set is a subset of every set. If our entire poset is the empty set, then there is no meet of the empty set in that poset, since there are no elements to be the meet.…
By the way, that makes a lot of sense despite perhaps being unintuitive at first. Every time you add an element to a set, the meet of that set gets “pushed farther down the poset”. So if the set is empty we need to start as far up as we can.
Simila…
I'll do meet. Following Definition 1.60 in the book, \(p\) is the meet of \(\emptyset\) if
for all \(a \in \emptyset\), \(p \leq a\), which imposes no restrictions at all, since there is no such \(a\);
for all \(q\) such that \(q \leq a\) for all …
Here's a start on the thrilling Puzzle 18. \(f_\ast\) does not always have a left adjoint, and here's an example.
Let \(X = \{1,2,3\}\) and \(Y=\{A,B\}\) and define \(f\) by \(f(1) = A, f(2) = A, f(3) = B\). \(f_\ast(\{1\}) = \{A\}\), so if the lef…
Puzzle 17. We must show that if \(S \subseteq T\) then \(f(S) \subseteq f(T)\), which is equivalent to saying that \(s \in f_\ast(S) \rightarrow s \in f_*(T)\). But if \(s \in f_\ast(S)\) then \(s = f(x)\) for some \(x \in S\), which means \(x \in T…
Thanks, it was enlightening to see the material from a slightly different angle!
[Removed the identification of a couple of typos that John has since fixed.]
OK, here's a pattern I don't think has been noted explicitly: we can get the list of reciprocals of the sequence (rather than the sequence itself) by reading each row right to left instead of left to right.
I'm doing all the exercises (through 1.80 so far) although I haven't talked about them here. They are excellent for ensuring that I understand the material before moving on but I think that they may be too numerous and too simple (in general) to clo…
Unfortunately the book does not make it clear that \(a \rightarrow c\) in Jerry's example is actually prohibited in a Hasse diagram (according to Wikipedia and Wolfram MathWorld, at least), not just redundant. I'm grateful to this example for spurri…
Let's see if I'm thinking along the same lines as John...
Puzzle 8: \(x \leq y \rightarrow x \cong y\). That is, the only way for elements to be related is to be in the same equivalence class.
Puzzle 9: Partitions.
Scott Fleischman: they do say, at the top of p. 11, "Contrary to the definition we've chosen, the term poset frequently is used to mean partially ordered set, rather than preordered set", but their wording makes it sound like it's more of a conventi…
Cole@38 + Joseph@45: Yes, I realized the same thing about terminal positions after sleeping on it. More trickily, we also would run into problems with forced captures (e.g., it's the only way to answer a check, or artificial "almost-stalemate" posit…
I don't believe that the set of all legal chess positions, with \(\leq\) meaning "can produce by legal moves", is a preorder set, since it does not satisfy the reflexivity condition. For example, if the current position \(x\) is a check in which the…
Puzzles 1-3 were useful in clarifying my understanding (thanks John) but I will leave the answers to others. For Puzzle 4, my go-to poset example is always the set of pairs \( (x,y) \in \mathbb R^2 \) where \( a \leq b \) when \( a_x \leq b_x \) and…