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Jun
18
comment Contraposition and law of excluded middle
@Materialist as far as I know (I am not an expert), the law of the excluded middle is formally equivalent to negation elimination. If we can obtain a statement without using negation elimination (i.e. without invoking $\neg\neg P \iff P$), we have proven it without making use of the law of the excluded middle. Here, we begin with, say, $(A\to B) \& \neg B$. If we can now show $\neg A$, then we've shown that $(A\to B)\to (B\to A)$. Now, assume $A$; we show that this leads to $B\&\neg B$, which allows us to conclude $\neg A$ without making use of the law of the excluded middle.
Jun
18
comment Extended Monte Hall problem (Hallway)
We start with $n$ doors, and each door has $n$ doors behind it? The answer will be the same—you're not really changing the problem very much; you're just jiggling some parameters around a little bit. Now not switching wins when you select the right door and then select the right door, which happens with probability $1/n^2$. Switching wins when you select the wrong door, change to the right door, and then select the right second door, which happens with probability $\frac{(n-1)}{n}\times\frac{1}{n-1}\times\frac{1}{n}>1/n^2$. Have I interpreted your question correctly?
Jun
2
comment What does “radical cube zero” mean?
For what it's worth, in mathematics, a "dimension" is not something that one can "enter". The dimension of a (vector) space is (roughly) the number of coordinates that it takes to fully specify a point in that space. So whatever references you've found to radical cubes (and I don't know what a radical cube is), it is doubtful that it has anything to do with entering or exiting other dimensions.
Jun
2
comment solve the equation for x
Hello, I've changed your statement of the problem to LaTeX code, which I would recommend learning to get the most out of this site. Can you check to make sure that I've translated correctly?
Apr
15
comment Proof by contradiction and order of statements
@Amateur yep. I take it back.
Apr
15
comment Proof by contradiction and order of statements
@Amateur I'm not sure that it's correct to say that $\mathbb{R}$ is a closed interval, although you're certainly right that it's a closed set.
Apr
15
comment Proof by contradiction and order of statements
For your last note, just make a truth table and check.
Apr
15
comment applications of derivatives : maxima and minima
Do you have any thoughts on this? What would it mean if the derivative of a function at a point was $10100$? What would the function look like at that point? Why couldn't that be a maximum?
Apr
15
comment Problem with trigonometric equation
That problem looks pretty tough to solve analytically to me. Where did you come across it?
Apr
15
comment Question about the Continuum Hypothesis
The answer to "where did I go wrong" is that OP took "Clearly this is either true or false - there either exists such a set, or there does not exist such a set" to be true, and this demonstrates that that's not so obvious.
Apr
14
comment If $x^2 +xy =10$ then when $x=2$ what is $\frac{\mathrm dy}{\mathrm dx}$?
do you mean "when $x=2$"?
Apr
14
comment Probability Equation That I am missing here
"Don't ask me why but I just logically see that it is 50% so why?" That's a dangerous way to think about math.
Apr
13
comment Writing Series as a Telescoping Series
yes, I was mistaken, sorry :(. I realized immediately and deleted my comment.
Apr
13
comment Proof by induction valid or not?
@DavidK I know. But either way, it is never correct to assume that the formula works for all $n$.
Apr
13
comment Writing Series as a Telescoping Series
That's not $S_N$. It's $t_N$ where $S_N=\sum_{i=9}^N t_N$.
Apr
13
comment For any continuous function f(x), how can I split up the function and restrict the domain to find an inverse?
This is a pretty broad question. You might be better off coming up with one or a few simple examples of problems you want to be able to solve and trying to work through those, maybe with help from this community if you get stuck.
Apr
12
comment When are $\Delta x$, $\delta x$, $dx$, and $\text{đ}x$ exactly the same? When are they approximately the same?
when you write $\delta x$, do you mean $\partial x$?
Apr
11
comment Why do some series converge and others diverge?
Let's for the moment restrict our attention to series with nonnegative decreasing terms. I think about it kind of like this. There are two opposing "forces" pushing on the behaviour of the partial sums of this series. On the one hand, each of the partial sums must be at least as big as as the last one, since we are adding more and more terms. This pushes up on the partial sums. On the other hand, with each term, the increases in the partial sums are decreasing, since the terms are getting smaller. If the series converges, it is because this second force is somehow stronger than the first.
Mar
13
comment Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)
But this clearly does not necessarily hold for all $x\in (t_i - t_{i-1})$'s, so we need to show that there is an $x_i$ in each interval such that it holds. But that is exactly what the mean value theorem promises.
Mar
13
comment Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)
@TuckerRapu $f$ can be discontinuous. MVT is a very powerful theorem that you can use to prove all kinds of things about the behaviour of a differentiable function on an interval. As littleO explains, we can get to $g(b)-g(a)\approx \sum g'(x_i)(t_i - t_{i-1})$ for some $x_i$ in the interval without appealing the the mean value theorem. Now we want to tighten that $\approx$ to an $=$, that is, we want to write $g(b)-g(a) = \sum g'(x_i)(t_i - t_{i-1})$...