This is a question on what is written here. If $A_i$ is a sequence of sets, define$$\liminf_i A_i = \bigcup_{j = 1}^\infty \bigcap_{i = j}^\infty A_i, \quad \limsup_i A_i = \bigcap_{j = 1}^\infty \bigcup_{i = j}^\infty A_i.$$What is an example where $\liminf_i A_i \neq \limsup_i A_i$? Take ...

I need to find the number of 3 digit numbers without repetition (distinct digits). MY ATTEMPT: All 3-digit numbers:$100,101,102,103,.....,999$ (i.e. $1000$ numbers) But we need to exclude following types of numbers _11,_22,_33,..._99 type (8*9 numbers = 72 numbers because the first digit sh...

Some time ago the synonym solution-verification$\to$proof-verification was created. (The synonym itself was discussed here, I'd say that without reaching consensus. And to me also it does not seem entirely clear what is better choice for the master tag. But anyway, the synonym exists.) But after...

Prove that $$\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$$ I can't find counting interpretations for either of the sides. A hint of "if $S$ is a subset of $\{1, . . . , n\}$ and $S^\prime$ is its complement then $|S| + |S^\prime| = n$" was also given, but I still don't know how to begin.

There's another possibility that hasn't been mentioned in the OP, and the one I would prefer: kill it! I don't see subspaces as a tag that will ever be consistently used, and not something that has a concrete enough meaning to allow for an easy delineation of when it should be used. Should you a...

Im having some problems isolating $\sigma$ in this expression: \begin{equation} 2\eta/ \sigma=A_1(\sigma/\mu)^{n_1} exp(-C_1)+A_2(\sigma/\mu)^{n_2} (b/d)^m exp(-C_2) \end{equation} Maybe someone can help me! Thanks in advance.

$$x + \frac{5} {2} = \frac{1} {6}$$ $$6 ( x + \frac{5} {2} ) = 6 ( \frac{1}{ 6})$$ $$3 ( x + 5 ) = 1$$ $$3x + 15 = 1$$ $$3x = -14$$ Why is it that we do that?

I was reading Wikipedia and came across the equation $e^Tx^2 + \frac{1}{T}xy + \sin(T)z-2=0$ and Wikipedia says that it is not a polynomial equation in the four variables $x,y,z$ and $T$ over the rational numbers(since sine, exponentiation, and $\frac{1}{T}$ aren't polynomial functions). However,...

The tag variables has been created recently. (The list of new tags says that (variables) was created Sep 5 at 9:04. So most likely in this question.) At the moment, there are 6 questions with this tag. The tag-info is empty and I have difficulties to imagine consistent usage of this tag. So I su...