What are the k-collections described in ch. 8 of “An Introduction to the Analysis of Algorithms” by Sedgewick

In chapter eight of “An Introduction to the Analysis of Algorithms” by Sedgewick (1996 edition) the coupon collector problem is introduced on page 425.

My confusion is how to identify the k-collections. A k-collection is defined as: “to be a word that consists of k different letters with the last letter in the word being the only time that letter occurs

Exercise 8.6 of the book asks to find all the 2-collections and 3-collections in Table 8.1, where that table shows the configurations of 4 balls in 3 urns

If I give it a try, I’d say a 3-collection from Table 8.1 is 2213, where the last letter (number 3) occurring just at the end, but I’m pretty sure I”m wrong.

Can anybody help providing an example of a k-collection (2 or 3-collection) from Table 8.1? Thanks

Is it weird to write out my character’s introduction and read it like a script? [on hold]

So, started a L5R rpg and wrote out my character’s introduction and read it like a script. I felt so embarrassed, everyone was grinning/smirking at me and I don’t know if it’s because I’m a girl, I’m awkward, I’m adorable, or it’s just plain weird and something to smirk at. Not that’s it terrible, I’m just wondering if it’s really weird and abnormal? Would you feel grinny to if someone did that as an intro to their character? I was sooooo shy about it too, because it’s like 5 other guys and they were all grinning at me

Introduction

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Introduction

Showing that two mappings are continuous. Introduction to topology

I have a course were we are learning about topology and have to show in two different exercises that two different mappings are continuous. I think I have a solution. However, as these concepts a new to me, I would like to ask if this is indeed correct and if there is a better way to solve them.

The first problem: Let $ (X,d_X)$ and $ (Y,d_Y)$ be metric spaces and define $ d_{X\times Y}:(X\times Y)\times (X\times Y) \rightarrow \mathbb{R}^+_0$ by

$ d_{X\times Y}((x_1,y_1),(x_2,y_2))=max(d_X(x_1,x_2),d_Y(y_1,y_2))$

Then show the projection

$ p_X:X\times Y \rightarrow X, p_X(x,y)=x$

is continuous.

I would then solve this by showing that for every open set V in X then $ p_X^{-1}(V)$ is an open set in $ X\times Y$ . So in mathematical terms

$ V \subseteq X$ and

$ \forall x\in V, \exists \delta_V >0 : B^X_{\delta_V}(x)\subseteq V$ and $ B^X_{\delta_V}(x)=\{y\in X|d_X(x,y)<\delta_V\}$

We then need there to exist a $ \delta_{X\times Y}$ so that $ \forall (x,y)\in p_X^{-1}(V), \exists \delta_{X\times Y} >0 : B^{X\times Y}_{\delta_{X\times Y}}((x,y))\subseteq p_X^{-1}(V)$

Where $ p_X^{-1}(V)=V\times Y$ as the second coordinate in $ (x,y)$ is just dropped and can therefore be anything.

As every $ x_1 \in V$ has a value $ \delta_X$ we can now look at every $ (x_1,y)\in p_X^{-1}(V)$ , where it is the same element $ x_1$ as before, then $ B^{X\times Y}_{\delta_{X\times Y}}((x_1,y))\subseteq p_X^{-1}(V)$ if $ 0<\delta_{X\times Y} \leq \delta_X$ .

I conclude this as for $ B^{X\times Y}_{\delta_{X\times Y}}((x,y))\nsubseteq p_X^{-1}(V)$ then there has to be a point $ (x_2,y_2)$ where $ d_X(x_1,x_2)<\delta_{X\times Y}\leq\delta_X$ and $ x_2\notin V$ as every possible $ y_2$ is in $ p_X^{-1}(V)=V\times Y$ . But this is contradictory to what we know as $ \delta_X$ is chosen so every $ x_2$ where $ d_X(x1,x2)<\delta_X$ is in V.

We can therefore conclude that for every open subset of X, called $ V$ , then $ p_X^{-1}(V)$ is an open subset of $ X \times Y$ as $ \forall (x,y)\in p_X^{-1}(V), \exists \delta_{X\times Y} >0 : B^{X\times Y}_{\delta_{X\times Y}}((x,y))\subseteq p_X^{-1}(V)$ is fulfilled and the projection is therefore continuous.

The second problem is so similar that it is not currently added.

Introduction …

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