# Tagged Questions

For requesting, clarifying, and comparing definitions of mathematical terms.

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### Why is the average defined the way it is?

The average, i.e. arithmetic mean is defined as: $$\mu = \frac{\sum_{i=1}^{N}x_i}{N}$$ for a data set consisting of $N$ data points. When we have two data points, the definition makes intuitive sense -...
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### Perfect powers by Oblath's result

What do you mean by this statement? Obl\'ath proved that the only perfect powers all of whose digits are equal to a fixed one $a \neq 1$ in decimal representation are 4, 8 and 9. This is equivalent ...
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### formula for defining terms in a finite set

Suppose there's a finite set, $S$ of terms in $\mathbb{R}$ which have the property $P(x)$. Suppose we know how to define the maximum value of the set by the relation, $max(x)$. We also have the ...
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### Relation between open sentences and sets (conceptual question)

Hi I'm a college student getting into the more proof oriented side of math. I was reviewing Mathematical Proofs, A Transition to Advanced Mathematics 2nd edition and after thinking about chapters 1 ...
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### When 2 functions are equal?

Are 2 functions equal when they have same domain, same codomain and same law ? EXAMPLE 1 $f: \mathbb{R} \to \mathbb{R}$ $x \to x^2$ and $g: \mathbb{R} \to \mathbb{R^+_0}$ (set of positive ...
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### Stabilization of embedding?

In D. Freed's lecture notes he mentions "stabilization of embedding" in theorem 4.48. Does anyone know the definition? I can't find it online.
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### Is there a difference between arc-wise connectivity or path-wise connectivity?

When authors refer to arc-wise connectivity, do they mean path-wise connectivity? I am studying space filling curves and when reading books, I either come across the concept of arc-wise connectivity ...
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### For a graded poset, why do we only consider the characteristic polynomial defined in terms of $\mu$?

When we have a graded poset $P$ with $0$ and $1$ we can define the characteristic polynomial $f_P(t)$ of $P$: $$f_P(t)=\sum_{x\in P}\mu(0,x)t^{r(1)-r(x)}$$ However, given a poset, have two functions, ...
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### Confusion Regarding Munkres's Definition of Basis for a Topology

The definition of Basis for a Topology as given in Munkres's book is as follows, If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ (called basis ...
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### Why there is no value for $x$ if $|x| = -1$? [duplicate]

According to the definition of absolute value negative values are forbidden. But what if I tried to solve a equation and the final result came like this: $|x|=-1$ One can say there is no value for $x$...
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### What is the formal definition of repeated limit?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers. I tried to formulate the definition but succeeded only ...
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### Relations, Ordered Pairs, Naive set theory by Halmos

I quote: "Explicitly: a set R is a relation if each element of R is an ordered pair;" The question is: "what about the converse? is a set of ordered pairs could be considered a relation?"