You see this trace on your oscilloscope for an RC circuit. The vertical scale is 5volt/square and the horizontal scale is 50ms/square. What is the time constant of the RC circuit?
So when the voltage is at its peak (15 V), that's Vo. and if I take that as V(t), it's V at 20 ms (approximately). Substituting this:
The left side of the equation is V(t) so you need to read off the value of V at 20 ms. For example, if it's 13 volts, then you will solve
Yeah, that's what I did..I said that the V at 20 ms was 15 (the peak). Should I just pick another arbitrary time?
I think your mistake is that you did not reset t=0 at the point where the decreasing exponential starts. If you measure V_0 at the peak, you must set t=0 at that point.
I have never seen this formally defined, only explained in words and used. I think that it logically means that "a is an element of S" is logically equivalent to "Fa." Am I right?
I usually read the vertical pipe in these cases as "such that". In many texts, the other symbol for "such that", the colon, is used in its place. One reads S={x|Fx} as "S is the set of all x such that (for which) Fx is true". See http://mathworld.wolfram.com/SuchThat.html .
Suppose that S is defined to be the set of all X, such that P(x) is true. My question now is this. Can a frog be an element of set S, yes or no?
While I'm at it, is there a symbol for "the nth element of"? As in, "appears as the first element of an ordered pair."
There is indeed an error; I've presented the set builder notation of naive set theory, which is known to be inconsistent. In ZFC, it becomes the following:
And if you're combining set theory with other axioms, you also have to specify that all the appropriate things are of the type Set.
Certainly, via the recursion theorem. However, usually one moves beyond the usual model of an ordered pair, and defines an n-tuple as a function whose domain is n. (That is, the set {0, 1, 2, ..., n-1}) So, asking for the "m-th element" is merely evaluating the function at m. This generalizes to using any set as the index set.
What actually prompted my question was the definition of a function f from A to B -- the set of all ordered pairs (A,B) such that every element in A appears exactly once as the first component of an ordered pair in f. So you can't use functions to describe the m-th element in that context. Order in an n-tuple must be more basic than the concept of a function, or else there must be a better way to define a function.
All one needs to define an ordered pair, is the set{{x},{x,y}} thus you can now define functions. If we assume the continuum hypothesis, then given any index set we can form an ordinal of the same set and talk about the s'th element in the set quite easily. However, forming products of indexed sets indexed over things other than subsets of the natural numbers isn't something you come across at a basic level, though it is important in Category Theory and such associated topics. For finite, positions, one can say that the m'th element of the ordered n-tuple is the element in the set of sets of cardinality m that is not in any of the set of sets of strictly smaller cardinality, where we assume cardinals of finite sets are well understood, since we could if we felt like it talk of them in terms of elements of the inductive set (assuming we're in ZF(C)).
Let me ask you something Hurkyl, if you don't mind. Suppose we let R denote a set, and let it denote the set of all sets which aren't elements of themselves. You are now asked to decide whether or not R is an element of itself. Using either ZFC or naive set theory (your choice), how would you make the decision?
Suppose we let R denote a set, and let it denote the set of all sets which aren't elements of themselves. You are now asked to decide whether or not R is an element of itself. Using either ZFC or naive set theory (your choice), how would you make the decision?
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