Vishal, same regarding vegetarian -> vegan. My issues with going full vegan are mostly social. I would be fine with changing my diet, but imagining going out to eat with people, or eating at people's houses, it seems much more difficult.
Just some mnemonics:
To remember the difference between \(\wedge\) and \(\vee\), I imagine the arrows in the Hasse diagram all pointing down, so the "smaller" elements are at the top. Then if \(a\) and \(b\) are arbitrary elements in the set, then …
The thing about your question is that we've been using \(\leq\) as the symbol for total orders, partial orders, and preorders. We also haven't really been using
The partial orders and preorders that are being discussed are not the order that you write the elements inside braces. Writing a set as a list of elements inside braces is not intended to imply an order. But the order you are talking about (letters …
Let \(X\) be a set, \(\mathcal{P}(X)\) its power set ordered by inclusion (\(A \leq B\) iff \(A \subseteq B\)), and \(\mathcal{P}(X)^{op}\) its power set ordered by containment (\(A \leq B\) iff \(B \subseteq A\)). Then the function \(\mathcal P(X) …
If \(X\) and \(Y\) are sets with preorders, you can define a preorder on \(X \sqcup Y\) by \(a \leq b\) iff \(a \leq b\) in \(X\) or \(a \leq b\) in \(Y\). This should be the coproduct in the category of posets. This is like putting \(X\) and \(Y\) …
What I've been doing for the past few days is just using the home button on the top left to see all discussions, or "Applied Category Theory Course" to see the Chapter and Lecture discussions.
I agree about the natural numbers including 0. I'm trying to think of how to justify the strength of my opinion on this though. Usually I think notation should be adjusted to the situation to facilitate explanation. Why is the definition of the natu…
Scott Fleischman, the difference between partial orders and preorders is that partial orders demand equality where preorders only demand equivalence. Equality is sometimes called "evil" in category theory.
https://ncatlab.org/nlab/show/principle+of…
Something I noticed when I learned category theory was that I found I was much more capable of holding conversations with mathematicians in very different fields than I was comfortable with, because I didn't necessarily need to know the fine details…
In a lot of math, the prefix "pre-" is added to a structure you want to talk about, but you don't quite have, but you have a general construction to get you there. Given a preorder, there is a general construction which turns it into a partial order…
I don't think trying to restrict the pieces so that every position can be altered and returned to in a positive number of steps is the right way to go. I think if you are in a checkmate position, then there is no way to return to that position in a …
Maybe I should have said "none of which are disconnected".
I've been chatting with my sister (who is home on spring break from UCSB, majoring in evolution and ecology) about ecology, and she's giving me ideas.