Recent content by mbs

Usually the definition of upper/lower bound would only imply s \leq \sup(S) for all s \in S and \inf(T) \leq t for all t \in T. In other words, the upper and lower bounds can be in the set themselves. The stated result should hold regardless though. Just start with \inf(T) \lt \sup(S) and go...

Okay. I figured it out.

Okay. One of my ideas was to construct finite sequences. Then the problem is showing that a finite sequence terminating in an empty set exists. If not, then a sequence of length ##n## must exist for every ##n \in \mathbb{N}##. I need to show that this is incompatible with (2) somehow. I...

Okay. I have the basic idea for the proof now. What I'm having trouble with is rigorously justifying it. I am considering sequences of subsets of ##A##. The sequences are defined inductively such that ##a_0## is ##A## itself, and if ##a_n## is a non-empty set in the sequence then ##a_{n+1}##...

I am trying to prove that two definitions of a finite set are equivalent. 1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## ) 2.) A set ##A## is finite if and only if...

Okay. I don't think I said anything that contradicts anything you're saying. A set for which the axiom of foundation fails is simply a (non-empty) set S in which every member of S contains another member of S. Simply removing the foundation axiom keeps such sets purely hypothetical. You would...

From my understanding, you don't have to define them. You simply omit the axiom of regularity/foundation and study the hypothetical "sets" that would violate it. \in is just a binary predicate relating two objects we happen to call "sets" in set theory. Every object in the domain of the...

Yes. It isn't that important because development of the integers and arithmetic do not depend on it at all. It simply forces the notion of a "set" to agree with what our intuition says a set should represent. Infinite or circular inclusion chains do not make a whole lot of practical sense.

Well, I live on the east side of Lake Michigan and thundersnow is still incredibly rare. Even when I've heard it, it typically only occurs once or twice throughout an entire storm. This in contrast to spring and summer thunderstorms which often produce hundreds or thousands of lightning flashes.

The "engine" that drives thunderstorm updrafts is latent heat release due to water vapor condensing into cloud drops. The Clausius–Clapeyron relation for water explains why water vapor can build to much higher concentrations in warm air than in cold air. In fact the relationship is...

I liked "Advanced Calculus" by Buck. It is fairly in-depth, uses a visual/intuitive approach to motivate proofs, and doesn't require a whole lot of background. It also has plenty of applications.

You have to look carefully at the definition of a relation. It should be something like... R\text{ is a relation iff... } \forall x : x\in R \Rightarrow x\text{ is an ordered pair}. If the predicate "x\in R" is false for every x, the statement "x\in R \Rightarrow x\text{ is an ordered pair}"...

I'm not familiar with that notation. Is it the same as the anti-symmetric tensor... \dfrac{\partial A_i}{\partial x_j} - \dfrac{\partial A_j}{\partial x_i} This is stuff I'm interested in learning at some point but haven't had the time.

Also want to point out any generalization of curl to 4 dimensions leads to a 6 dimensional value. But I'll first explain 3 dimensional curl. Three dimensional curl can be geometrically interpreted as a rate of rotation induced by a vector field. It can be described using a 3 by 3 matrix...

You just have to check that each solution satisfies the original constraint, and all four do. The two x=0 solutions are equal maximums and the two y=0 solutions are equal minimums. This is an easy problem to check visually if you can picture it.