Yes, this is a good point. Hmm. It looks like a to-my-mind more acceptable contrapositive formulation of the statement will not be valid in ZF. That's a pain. I guess in ZF, however, we can rely on ex-falso-quodlibet to say ##\forall x, x \notin \emptyset'## and then ##x \in \emptyset'...
I'm going to reanimate this thread briefly re: the "unique empty set" question.
The way I think of this is as follows: Suppose ##\emptyset## and ##\emptyset'## are both empty sets. We need to show that ##\emptyset \subseteq \emptyset', \emptyset' \subseteq \emptyset##.
For the first, we have...
OK, I guess that you are trying to stress that the x refers to an arbitrary element of A; typograhically however, there are two xs, and in a natural language translation of the notation, you can't put "any" in front of the second: "any x in B" doesn't make much sense.
To ensure that this thread doesn't derail, I ought to point out that my question is *purely* about notation. I do indeed have no confusion about the concept of a subset, but rather about the conventions used in used in logic (I guess this is first-order logic?) to notate them. I managed to get a...
What does the subscripted ##x \in A## notation mean? I've never seen that before.
I don't follow what point you are making here; and also, there are two ##x##s - I guess you are referring to the first?
1. I think that I probably should have written "predicate" not "proposition"
2. I'm not sufficiently familiar with logical notation to know what the conventions are, I'm afraid.
3. I note that this wikipedia section on quantifiers suggests that ##\forall x \in A (x \in B) \Leftrightarrow \forall...
Slightly short of time right now but one question:
##x \in B## is a proposition, isn't it? So does ##\forall x \in A(x \in B)## not say "for all x in A, the proposition x is in B is true"?
OK, thanks. That is how I would have interpreted it.
That's true, but it betrays the way I tend to think about the question: how would you automate the procedure in a programming language, say, or how would you manually check the subset status of two sets written in list notation, maybe.
To my...
This is a somewhat trivial question, but I never managed to learn much logic back in the day...so:
The definition of a subset can be written as:
## A \subset B \Leftrightarrow \forall x (x \in A \Rightarrow x \in B) ##
However, over which set is ##\forall## supposed to quantify? It seems to...
I don't have time to reply to this fully today, but your final observation suggests that I've indeed effed up somewhere.
I'll put up the details tomorrow.
Well, nothing too unusual: ##Q=\dfrac{\omega_\text{resonance}}{B_\text{3dB}}## where ##B_\text{3dB}## is the 3dB bandwidth.
I find your question a little odd though; aren't all definitions equivalent up to a factor of ##2\pi##. It's been years since I looked at this stuff in detail, but I don't...
I've been experimenting with an LC tank circuit in series with a resistance R, and I've noted that the Q seems to increase with R. I've tried to derive this result via phasor analysis, but I'm not sure if my expression is correct.
To make things clear, I'm talking about the circuit with...
OK, it's clear that this is trivial. Thanks. In fact, the only relevant text I have available (Streetman, Solid State Electronic Devices) does in fact treat this topic but without using the name Boltzmann anywhere - I skimmed the section in question without really noticing the result, which...
OK, thanks, things are slightly clearer.
However, I'm not sure either what your ##N## or the expression on the RHS are. Is ##N## related to the density of states (i.e. what I'd call ##N(E)##), or is it the concentration of charge carriers? And the expression on the RHS looks suspiciously like...