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8

A binary relation (or relation, means the same) from a set $A$ to a set $B$ is any subset $R\subseteq A\times B$. We take any here seriously so in particular, if $A$ contains some element $a$, and $B$ contains some element $b$, then $R=\{(a,b)\}$, being a subset of $A\times B$ is a relation from $A$ to $B$. For that matter, given any two sets $A$ and $B$, ...


7

$$ \left( \sum_{k=1}^{n} \dfrac{1}{k} - \ln(n+1) \right) - \left( \sum_{k=1}^{n} \dfrac{1}{k} - \ln(n) \right)=\ln(n)-\ln(n+1)=\ln\left(\frac{n}{n+1}\right)$$ And $$\lim_n \ln\left(\frac{n}{n+1}\right)=\ln 1=0$$


6

$n^k = \underbrace{n\cdot n \cdots n}_{k\ \textrm{times}}$ only works when $k$ is a positive integer. In order to define exponentiation more generally, we must refine the definition of exponentiation several times: First, we must address a suitable definition for $k = 0$ and $k \in \{-1, -2, \ldots \}$, including operations of the form $n^{k_1}n^{k_2}$ ...


4

Because only multiplication of two numbers is defined at that point. Also, if it's not yet defined to be associative, that expression may be ambiguous.


4

IF the collection $\mathcal G$ of all finite groups $(G,\times)$ is a set, then you can define a set: $$\mathcal S = \bigcup_{(G,\times)\in \mathcal G} G$$ In particular, if $Y$ is any set, there is a group on the singleton set $G=\{Y\}$. So for any set $Y$, $Y\in \mathcal S$. That means $\mathcal S$ contains all sets. That is not allowed in set theory ...


4

just trust the definition! a relation on a set $A$ is any element of $\mathfrak{P}(A \times A)$. a singleton is such an element. don't confuse with a function $A \to A$, which is a relation which must satisfy two further conditions. what about the empty relation?


4

The reason we have charts on a topological manifold is so that if we pick a chart $U \to \mathbb{R}^n$ that is a continuous function, we can say that the topological space $U$ is homeomorphic to the topological space $\mathbb{R}^n$, so any questions about the topology of $U$ can be moved to questions about the topology of Euclidean space, which we understand ...


3

It is the slice category (a special case of comma categories), sometimes also denoted by $\,1\!\downarrow {\bf Set}$, its objects are the arrows $1\to A$ of ${\bf Set}$ and its morphisms between $a:1\to A$ and $b:1\to B$ are arrows $f:A\to B$ that makes the triangle commutative, i.e. satisfying $\ f\circ a=b$.


3

Here's a neat way to think about it, I think. A vector space is a mathematical structure that can be built as follows. We start with a set of objects that, first of all, you are able to add/subtract together (or, in the language of algebra, an abelian group). As long as you are able to add things, it makes sense to define, at the very least, integer ...


3

Typically: $R[x]$ denotes the set of polynomials over $R$ When $R$ is a domain, $R(x)$ denotes the set of rational polynomials over $R$ $R[[x]]$ denotes the formal power series over $R$ $R((x))$ denotes the Laurent series over $R$ vuur asked an interesting question in the comments which I can speak to here. The answer is "If $R$ is a commutative ...


3

As @N.S. pointed out, it does not make any difference in the limit by choosing $\ln(n+A)$ rather than $\ln(n)$. The use of $\ln(n)$ comes from the convenient integral formula given by $$\gamma = \int_1^\infty \left( \frac{1}{[x]} - \frac1x \right) dx.$$ Here the choice to use $\ln(n)$ is natural. Moreover, we can view $\gamma$ as the difference in the ...


2

Here is the answer that I wrote a few months ago, which should pertain to this question as well: Both categories are denoted by $\mathrm{Rel}$. In most cases, however, the author either explicitly tells which $\mathrm{Rel}$ he/she is using, or reproduces the definition of $\mathrm{Rel}$. (Note: this applies to any definition/term in math which can be ...


2

The standard definition of a class of groups is a a class $\mathcal{C}$ (in the set-theoretic sense) whose members are groups with the property that if $G$ and $H$ are isomorphic groups then $G \in \mathcal{C} \Leftrightarrow H \in \mathcal{C}$. So yes, there is a class of finite cyclic groups and a class of nilpotent groups, etc. It is customary to study ...


2

Yes, every set can be made into a subset of something that has a vector space structure. But that's not really the important thing here. What the definition means is that you you say "here is a functional", then you're implicitly promising to tell me not only what the map you're thinking of, but also which vector space you're thinking of the domain as a ...


2

This was going to be a comment but it does not quite fit. It sound like your question has something to do with bridging the divide between an intuitive or "picture" understanding of homology and the formal definition of singular or simplicial homology in the language of chain complexes. You can of course define homology without using the language of ...


2

Why do we write $ab$ instead of $$\underbrace{b+b+\cdots+b}_{a \textrm{ times}}~~~?$$


2

If $(A,\leq)$ is a partial order, then we define these two definitions for $a\in A$: $a$ is maximal if whenever $a\leq b$, then $a=b$. $a$ is maximum if for every $b\in A$, $b\leq a$. You can prove that every maximum is maximal, but a maximal element need not be a maximum. In particular there can be many maximal elements. So being maximal and maximum are ...


2

The links below are to the Mathematical Atlas. Most of these fall under the large area called analysis. 1) and 4) are (surprisingly?) differential equations. Maybe someone else can provide more insight than I can? As for div and curl, these can be studied in more generality in differential geometry. 2) The Laplace transform is a technique that is used both ...


2

Regarding Edmund Landau, Foundations of analysis : The arithmetic of whole rational, irrational, and complex numbers (ed or : 1930), its approach is the "standard" axiomatic one. See pages 1-2 : We assume the following to be given : A set (i.e. totality) of objects called natural numbers, possessing the properties — called axioms — to be listed below. ...


2

What constitutes the definition of a function? Must not the domain always be specified clearly, unambiguously, distinctly and separately before making use of the said function for something else? Yes. f is said to be a function (of one variable) mapping all elements of set A to elements of set B iff $f\subset A\times B $ $\forall x\in A: \exists ...


1

Here notice that $X\cap Y=X$ because $X\subset Y$ since $2$ is even. So generally when events $A$ and $B$ and $A\subseteq B$ then we have $P(A\cap B)=P(A)$. Also since $P(A\cap B)=P(A|B)P(B)$ so if $P(A|B)=1$ you have $P(A\cap B)=P(B)$ but also $P(A|B)=1$ will generally be true when $A\subseteq B$


1

Uncountable additivity are considered especially in set theory. For example, an uncountable cardinal $\kappa$ is a real-valued measurable cardinal if and only if there exists a nontrivial $\kappa$-additive probability measure on $\kappa$. Although in some cases $\sigma$-additivity is enough. For example, the least cardinal which has a $\sigma$-additive ...


1

I'm sure this example will solve your problem: $$f(z) = |z|^{2}\text{.}$$ Considering this one we can clearly see that this is only differentiable at $0$ while the criteria for analyticity is that you have some $k>0$ such that this still stays differentiable. Hence differentiability does not imply analyticity.


1

Unless I've misunderstood what you've written, there are no such non-zero functions $f$. E.g. if $M = \mathbb R$, then you're asking for functions $f: \mathbb R \to \mathbb R$ such that $f\circ \phi^{-1}$ is smooth for all homeomorphisms $\phi$ of $\mathbb R$; the only such functions $f$ are constant functions. (Note that, replacing $\phi$ by $\phi^{-1}$ ...


1

Let's define the ordered sum of a poset-indexed family of posets! The ordered sum of such a family $J = \{J(i)\}_{i \in I}$, which I shall denote by either $\bigoplus J$ or $\bigoplus_{i \in I} J(i)$, is the poset whose underlying set is $$\bigoplus_{i \in I}J(i) = \{(i ,j)\mid i \in I \text{ and } j \in J(i)\}$$ and whose ordering $\le_{\oplus J}$ is ...


1

The problem is that $\arg(z)=\arctan\left(\frac{\Im(z)}{\Re(z)}\right)$ is only true for $\Re(z)>0$ Here's the full definition $$\arg(x+iy)=\left\{ \begin{array}{lr} \arctan\frac yx&\quad x>0\\ \pi+\arctan\frac yx&\quad y\ge0,x<0\\ -\pi+\arctan\frac yx&\quad y<0,x<0\\ \frac\pi2&\quad y>0,x=0\\ -\frac\pi2&\quad ...


1

This really isn't a question specific to arguments and complex numbers. Essentially the arctan function horizontally asymptotes to $pi/2$ and $-pi/2$, as seen by the graph: So there actually isn't a value of $x$ (infinity is not a number!) that gives an output of $-pi/2$. Rather, the limit as $x$ approaches negative infinity for $arctan(x$) yields the ...


1

Although this is hardly the reason that cause confusion because people have common sense, "..." is not well defined if you want to be picky. For example $ \{1,2,3,4,...,100\} $ could mean set of natural number less or equal to 100 or the image of that set under function $f(n)=(n-1)(n-2)(n-3)(n-4)(n-100)+n$ Define things using recursive remove the ...


1

Most certainly a relative notion. You define the neighborhood $N_r(x)$ of $x$ in the metric space $X$. So, in particular, a neighborhood in the Euclidean space $\mathbb R$ is very different than a neighborhood in the subspace $\mathbb Q$. The former contains uncountably many elements, while the latter only countably many, and all are rational numbers. It ...


1

I don't know specifically in Rudin's book how these notations are used, but in general one would want to keep the notion as flexible as possible, and there would be many occasions where one would want to view $A$ as a metric space in its own right, along with all of the associated notions such as that of an $r$-neighbourhood. So the notion of a relative ...



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