# Tagged Questions

For questions about the motivation behind mathematical concepts and results. These are often "why" questions.

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### Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
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### Motivation for the study of algebraic structures

I am currently studying group theory and I realized that most concepts we study are just definitions on which we build theory. I do understand that some theorems are beautiful and don't need any ...
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### Why is Grobner basis useful? [duplicate]

I don't get the motivation for calculating Grobner bases. What's good by computing a Grobner basis for an ideal of $k[X_1,...,X_n]$? Moreover, is there any theorem whose proof relies on the use of ...
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### Motivation for introducing $A^{\prime} := \frac{1}{\left|G \right|} \sum_{g \in G} \Delta(g^{-1}) A \Delta(g)$ in representation theory proof

Assume $\left|G \right|= \left| G \right| \cdot 1_F$ is invertible in $F$. Let $\Delta:F \to GL_n(F)$ be a representation and $U \subseteq F^n$ be an $F$-subspace that is $\Delta$-invariant. Then ...
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### Why was Sheaf cohomology invented?

Sheaf cohomology was first introduced into algebraic geometry by Serre. He used Čech cohomology to define sheaf cohomology. Grothendieck then later gave a more abstract definition of the right derived ...
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### What is the 'meaning' of nowhere dense set?

In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres). So what is the 'meaning' (i.e motivation, ...
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### Providing motivation for the importance of the concept of a 'basis'.

In a few situations, I found myself being asked by younger students why the concepts of a basis was important. First, in the concept of linear spaces, it's easy to explain that having a basis allows ...
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### The (matrix) definition of a positive-definite function

In the definition of a positive-definite function, why are the elements of the n$\times$n matrix chosen as $f(x_i-x_j)$ for $i, j = 1,...,n$? Also, it says that "for any real numbers $x_i$". Does ...
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### Motivate why $a^{-n}$ equals to $\frac{1}{a^n}$

I have to prove that $a^{-n} = \frac{1}{a^n}$ with $\frac{3^4}{3^7}$, but before I can do that I have to understand the background. The background says: we know that $\frac{3^4}{3^7} = \frac{1}{3^3}$...
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### Banach spaces containing copies of $\ell^1$

Why is it important that a Banach spaces $X$ contains (or not) copies of the space $\ell^1$? I always hear talk about it but I don't know its importance. Could someone explain this?
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### Equal Categories [closed]

Let $\infty$ be the "category" of all categories, where the objects are categories and the morphisms are functors. I am trying to motivate the definition of equal categories by doing it as follows. ...
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### What motivates the definition of a ring in abstract algebra? [closed]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for theorems?...
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### Motivation for the study of units in cyclotomic fields beyond Washington's book

Right now, I am reading Larry Washington's book "Introduction to Cyclotomic Fields." In Chapter 8 of this book, the unit group of the ring of integers in a cyclotomic field (or its totally real ...
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### Motivation for separation axioms

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am ...
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### Justification for ordinal arithmetic at limit ordinals

Wikipedia tells me that: $$\alpha + \lambda := \bigcup_{\beta < \lambda} \left ( \alpha + \beta \right )$$ for a limit ordinal $\lambda$. Multiplication and exponentiation, as Wikipedia says, ...
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### Ordinals - motivation and rigor at the same time

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal ...
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### Traveling salesman problem: why visit each city only once?

According to wikipedia, the Traveling Salesman Problem (TSP) is: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city ...
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### How to understand cocategories

$\newcommand\CC{\mathsf{C}}$The notion of a category is well-known. There are multiple equivalent definitions; small categories can be seen as an internal category in $\mathsf{Set}$, that is a ...
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### What is the motivation to continuous functions and measurable functions?

In topology the objects of interest are the space open sets, and a function will be continuous if the inverse image of any open set is an open set. In measure theory the objects of interest are the ...