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location Cleveland Heights, OH
age 66
visits member for 3 years, 3 months
seen Jan 2 at 12:05

Professor emeritus at Cleveland State University. I’m a set-theoretic and general topologist with an interest in combinatorics. I’m also interested in linguistics, especially historical linguistics.


May
15
comment Confused about combinatorials
@Nick: Let me know if that helps.
May
15
comment Is every set that's a union of two sets disconnected?
@Roger: See if this helps. $[0,1)\cup(1,2]$ has a natural break between $[0,1)$ and $(1,2]$. For instance, no sequence in $[0,1)$ can converge to any point in $(1,2]$, and vice versa. You can’t split $[0,2]$ that way: no matter how you split it into two non-empty sets, there are sequences in one set that converge to points in the other, so the two sets are somehow tied together.
May
15
revised Confused about combinatorials
added 297 characters in body
May
15
comment Confused about combinatorials
@Nick: Okay, so you’re studying very elementary combinatorics. In that context I’ll modify the answer a little; it shouldn’t take more than a few minutes.
May
15
comment Is every set that's a union of two sets disconnected?
@Roger: The sets $[0,1)$ and $(1,2]$ are relatively closed subsets of $[0,1)\cup(1,2]$. Indirectly, that’s why $[0,1)\cup(1,2]$ is not connected: if they were relatively closed, then, since they’re complementary in $[0,1)\cup(1,2]$, they’d also be relatively open and hence constitute a separation of $[0,1)\cup(1,2]$. But that’s not the kind of answer for which I was looking. I’m asking whether it makes any kind of intuitive sense to you that $[0,1)\cup(1,2]$ is not connected, while $[0,2]$ is connected.
May
15
comment Confused about combinatorials
@Nick: It really depends on just how one interprets combinatorial proof. This argument is borderline. Give me some context: what sort of material are you studying? I may be able to make it a bit more obviously combinatorial.
May
15
comment Is every set that's a union of two sets disconnected?
@Roger: That’s actually just the usual definition, so there’s nothing to prove unless you’ve been given some other definition as a starting point. // Do you have any intuitive sense of why we’d want to consider $[0,2]$ connected but $[0,1)\cup(1,2]$ disconnected?
May
15
comment Confused about combinatorials
@Nick: Yes, that’s right. The rest of the argument is that the area of a rectangle is the product of its side dimensions, and by actual count of unit squares these rectangles have the same area, so ...
May
15
answered Confused about combinatorials
May
15
answered Is every set that's a union of two sets disconnected?
May
14
revised Solving two algebraic equations
Better LaTeX.
May
14
comment Cantor set as a set of continued fractions?
I don’t think that it actually helps: it just uses the result that $C+C=[0,2]$, where $C$ is the middle-thirds Cantor set, as motivation for a similar result about continued fractions with bounded denominators.
May
14
answered Show using induction (coupled linear recurrences)
May
14
revised Show using induction (coupled linear recurrences)
Copied image.
May
14
comment Should $\mathbb{N}$ contain $0$?
I’ve voted to close this as not being constructive. Although the Question is formally not an invitation to argument, it really does just solicit opinions. Worse, those opinions mean virtually nothing, since people will in general use their preferred notation regardless of the views of others.
May
14
comment Intersection of collection of sets not in the collection of sets?
@Roger: You’re welcome! I deliberately used a variety of notations in hopes that it would help.
May
14
answered Intersection of collection of sets not in the collection of sets?
May
14
answered How to show that$(X,d)$ is totally bounded
May
14
answered Stirling Numbers Combinatorics
May
14
answered Number of ways to arrange $n$ people in a line