## Does |xy| ≤ p in the pumping lemma count for all i?

While learning about the pumping lemma, I came across the following question:

Given the language L is $$a^n(0|1)^*$$ with $$c_0 \cdot c_1 = n$$, where $$c_0$$ indicates the amount of zeros present, use the pumping lemma to prove that this language is not regular. Examples of valid words in L are 0, 1, a01, aa001, etc…

In regular english: a word with leading as, followed by any amount of either 0 or 1 characters, where the amount of zeros multiplied by the amount of ones needs to match the amount of as.

My initial attempt was the following:

• Pick w as $$a^p0^p1$$. This obviously holds since $$p \cdot 1 = p$$.
• Split w into xyz as $$x = \epsilon$$, $$y = a^p$$, $$z = 0^p1$$.
• Show that $$xy^2z = a^{2p}0^p1$$ does not hold, since $$p \cdot 1 \ne 2p$$.

However, the answer key instead opts to introduce two new variables $$m \geq 0$$, $$n \gt 0$$, then splits on $$x = a^m$$, $$y = a^n$$, $$z = a^{p-n-m}0^p1$$ (splitting the sequence of a into three parts). Then, they too use $$i = 2$$ to show that $$xy^2z = a^{m}a^{2n}a^{p-n-m}0^p1 = a^{p+n}0^p1$$, which is not a member of the language (since n was more than 0).

As far as I can see, my attempt adheres to the $$|xy| \leq p$$ condition of the pumping lemma: x is empty and $$|y| = p$$. As such, I was under the impression that my answer was correct.

However, the huge increase in complexity of the answer in the answer key leads me to believe that this is not a valid approach. I have the sneaking suspicion this is because my answer actually does violate the $$|xy| \leq p$$ condition.

Is my attempt a correct way to prove that the language is not regular? If not, what misconception/mistake did I make along the way?

## Find PDA for CFL = {x#y | |x| = |y| and x $\neq$ y}

I am studying push down automata. When I read a solution for show that CFL L = {x#y | $$x \neq y$$, $$x,y \in \{0,1\}^*$$}, I could understand L = $$L_1 \cup L_2$$, $$L_1 = \{x\#y\enspace|\enspace|x| \neq |y|\enspace\}$$, $$L_2 = \{x\#y \enspace|\enspace|x| = |y|$$, $$x \neq y\enspace\}$$, and make CFG for $$L_1$$, but I don’t know how to make CFG for $$L_2$$, Since y is not a reverse string of x. How can I know $$i_{th}$$ symbol is different?

## Find numbers $x$, $y$ such than $(x,y) = 18$ and $[x,y]= 3780$

Where $$(,)$$ is the greater common divisor and $$[,]$$ the smallest common multiple.

In general how do you solve this kind of problems?

## If $X$ is a vector space endowed with a metric that is NOT norm induced and If $x_n\to x$ and $y_n\to y$, then $x_n+y_n\to x+y$

Let $$X$$ be a real or complex vector space endowed with a metric $$d$$ which is not induced by a norm, that is there exist no norm $$\| \cdot \|$$ on $$X$$ such that $$d(x,y)=\| x-y \|$$. Prove or disprove:

(a) If $$x_n\to x$$ and $$\lambda_n\to \lambda$$ ($$\lambda_n,\lambda$$ are scalars), then $$\lambda_n x_n \to \lambda x$$.

(b) If $$x_n\to x$$ and $$y_n\to y$$, then $$x_n+y_n\to x+y$$.

This is throwing me off just because the vector space is endowed with a metric that is not norm induced, and I am not sure what all that implies. With that said, I think I may be doing this problem naively. Anyway, here are my solutions:

(a) Let $$y\in X$$ be fixed. Then, by the triangle inequality, $$d(\lambda_nx_n,y) \leq d(\lambda_nx_n,\lambda x) + d(\lambda x, y),$$ similarly, $$d(\lambda x,y)\leq d(\lambda x,\lambda_n x_n) + d(\lambda_n x_n , y),$$ which implies that $$d(\lambda x,y) – d(\lambda_n x_n , y) \leq d(\lambda x,\lambda_n x_n),$$ so that as $$n\to\infty$$ we have $$\lambda_n x_n \to \lambda x$$.

(b) We may establish an upper bound via the triangle inequality, so that $$d(x_{n},y_{n}) \leq d(x_{n},x) + d(x,y) + d(y,y_{n}).$$ Again for a lower bound, we use the triangle inequality: $$d(x,y) \leq d(x_{n},x) + d(x_{n},y_{n}) + d(y_{n},y);$$ hence we have that, $$d(x,y) – d(x_{n},x) – d(y_{n},y) \leq d(x_{n},y_{n}).$$ Now as $$n \to \infty$$ we have $$d(x_{n},y_{n}) \to d(x,y).$$

## How many pairs $(x,y)$ such that $x+2=y$ have an $x$ or $y$ divisible by 5 and not 3?

Let the set $$S_{n}$$ = {$$(x,y):x,y \in \mathbb{O}$$} such that $$x+2=y$$ and $$y$$ is less than or equal to odd integer $$n$$ and $$\mathbb{O}$$ is set of odd integers > 1.

Let us define the function $$f(n) = |S_{n}|$$ that counts the number of pairs in $$S_{n}$$.

Examples: $$S_{9} =\{(3,5),(5,7),(7,9)\}$$ and $$f(9) = 3$$.

$$S_{13} = \{(3,5), (5,7), (7,9), (9,11),(11,13)\}$$ and $$f(13) = 5$$

$$S_{35} = \{(3,5), (5,7), (7,9), (9,11), (11,13), (13,15), (15,17), (17,19), (19,21), (21,23), (23,25), (25,27), (27,29), (29,31),(31,33),(33,35)\}$$

and $$f(35) = 16$$

It turns out that $$f(n) = ((n-3)/2)$$ for odd integers $$n$$.

Let us define another function $$g(n)$$ that counts the number of pairs $$(x,y)$$ such that either $$x$$ or $$y$$ is divisible by 5 and not divisible by 3, and $$x \neq 5$$ and $$y\neq 5$$.

Example: $$g(35) = 3$$ because there are 3 pairs where $$x$$ or $$y$$ is divisible by 5 and not 3, and $$x$$ or $$y$$ is not equal to 5. They are (23,25), (25,27) and (33,35).

Note, the pairs (3,5) and (5,7) are not included since they contain 5, and the pairs (13,15) and (15,17) are not included because 15 is also divisible by 3.

What is the formula for $$g(n)$$ in terms of $$f(n)$$ ?

What is the formula for $$g(n)$$ in terms of $$f(n)$$ for limit $$n \to\infty$$?

## Random variables $X,Y$ such that $E(X|Y)=E(Y|X)$ a.s.

Let $$X$$ and $$Y$$ be random variables such that $$E(|X|), E(|Y|)<\infty$$ and $$E(X|Y)=E(Y|X)$$ a.s. Then is it true that $$X=Y$$ a.s. ? If this is not true in general, what happens if we also assume $$X,Y$$ are identically distributed ?

## $X,Y$ spaces such that are cones over $X\cap Y$ generate $X\cup Y=\sum (X\cap Y)$

I have this question from this Kratzer and Thevenaz article :

In page 89, they explain that

If $$Y=Y_1\cup Y_2$$ are such that $$X=Y_1\cap Y_2$$ and if each $$Y_i$$ is a cone over $$X$$, then $$Y$$ is the suspension of $$X$$

But in page 90, in the proof of the Proposition 2.5, they use this in a way I don’t understand. They prove that $$\mid X\mid \simeq C\mid E\mid$$ and $$\mid Y\mid \simeq C\mid F\mid$$, but those aren’t cones over $$\mid X\cap Y\mid$$, so why I can continue the proof of this proposition saying that this implications make $$\mid G\mid \simeq \sum(\mid X\cap Y\mid)$$ ?

My concept of a “cone over $$\mid X\cap Y\mid$$” is:

$$\mid X\cap Y\mid\times [0,1]/ (a,i)\thicksim(b,1)$$

with $$a,b\in X\cap Y$$.

Am I misunderstanding what is a cone over $$\mid X\cap Y\mid$$?