# Tagged Questions

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

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### n-tuple function definition

I've read this definition for an hour now and I cannot piece it together abstractly. To define an n-tuple as a function $F$, where $X$ is the index set and domain, and $Y$ is the set containing the ...
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### How many teams of $5$ players out of $15$ girls and $10$ boys can be formed with at least $2$ boys and $2$ girls [with complement]

How many teams of $5$ players out of 15 girls and 10 boys can be formed with at least 2 boys and 2 girls? The solution has to be with complement. This is related to: How many ways to assemble a team ...
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### Definition of the tensor product of finite sequence of modules

I have posted several questions about the tensor product of modules before and this post would be the final one. I have read wikipedia,mathSE,Dummit&Foote and Bourbaki for the definition of the ...
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### What is the difference between an antiderivative and an integral?

In my textbook, it states the fundamental theorem of calculus as follows: If $f(z)$ has an antiderivative $F(z)$, then $\int^{z_2}_{z_1} f(z)dz=F(z_2)-F(z_1)$. There isn't a definition of what an ...
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### Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...
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### Use of the word “if” in mathematical definitions [duplicate]

I'm looking at the following definition The random variables $X_{1}, \ldots, X_{d}$ are said to be comonotonic if they admit as copula the Frechet upper bound. I am however not quite sure how to ...
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### Is free abelian group a “free” abelian group or “free abelian group”?

Let $G$ be an abelian group. What does it mean that $G$ is a free abelian group? Does this mean that $G$ is a free group or a free-$\mathbb{Z}$-module with the operation $n•a=a+...+a (n-times)$? Or ...
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### Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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### Dirac Delta Function definition with ksi (ξ)

The dirac delta function has a definition $$f(0)=\int_∞^∞f(x)δ(x)dx$$ and $$f(x)=\int_∞^∞f(x-ξ)δ(ξ)dξ$$ (the lower bound is minus infinity but I couldn't add a minus :/) I do understand the ...
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### If for any $\varepsilon$ exists $\delta$, does that mean that for every $\delta$ exists $\varepsilon$? [closed]

For any $\varepsilon \gt 0$ exists $\delta \gt 0$, does that mean that for any $\delta \gt 0$ exists $\varepsilon \gt 0$? If $\delta$ depends on $\varepsilon$ such as $\delta = \frac 1 \varepsilon$, ...
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### Why use open sets in definitions?

I've been wondering why do we use mostly open sets in mathematics to define numerous things. For example continuity is defined using (open) neighbourhoods, differentiability and Taylor's theorem is ...
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### What do these definitions of conjugacy have in common?

Here are four (seemingly) different uses of the word conjugate: Complex conjugates are a concrete instance of the idea of conjugacy in field extensions. In group theory, there's the idea of ...
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### Memorising lots of maths theorems/lemmas

In a few weeks I'll have my summer exams in a Senior Freshman Mathematics course. Two of my modules have a huge number of theorems, lemmas and definitions - over 200 definitions and 250 proofs by my ...
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### What's the difference between list and sequence (as mathematical concepts not programming point of view)?

What distinguishes between set, multiset and list is whether the order is important or not, and whether repetitions of elements is allowed: List: order is important and repetition is allowed ...
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### Definition of argument-based $\min$ and $\max$

When I say "argument-based", I mean $\min$ and $\max$ functions that take arguments, as opposed to the min and max of a function. For example: $\min(3,4,-2)=-2$, $\min(100,3)=3$, $\max(3,4,-2)=4$, etc....
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### Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
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### What is $\mathbb{Z}[a]$?

Let $\alpha$ be algebraic over a field $F$. Then, $F(\alpha)$ denotes the subfield of $F$ generated by $\alpha$. This is the standard definition of $F(\alpha)$. Under this definition, for example, ...
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### What is the justification for calling a hereditary system an independent system?

I was learning about set systems and hereditary systems and I noticed that they also call a hereditary system a independence system and that didn't quite make sense to me intuitively. First recall ...
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### Difference between transform and transformation.

I was told that there is a difference between a transform and a transformation. Can anyone point out clearly. For example : Is Laplace Transform not a transformation ?
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### Prove a limit using the formal definition of the limit

So I have a sequence {a_n} = π/2^n where n=1,2,3,4.... And I need to prove that its limit is 0. Here is what have done, can someone check and tell me if this is correct.? Definition: A sequence {...
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### What is $g^1_3$?

I'm trying to find the definition of $g^1_3$ in algebraic geometry Hartshorne's book, anyone who is used with this book could help me to find this definition? Thanks Remark: this extract is from ...
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### Why is it difficult to define n-category?

Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally. If we have a definition for 1-categories (category) and a definition for ...
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### Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
### why does we need to satisfy $d(x,y)\le d(x,z)+d(z,y)$ in order to show metric space?
In metric space some axioms must be satisfied . I wonder why we need to satisfy $d(x,y)\le d(x,z)+d(z,y)$ in order to be metric space. If this axiom is not satisfied, does any problems occur? ...
All textbooks and websites I've consulted define function composition thus: Let $f: A \rightarrow B$ and $g: B\rightarrow C$ be functions. The composite of $f$ and $g$ is the function \$f \circ g: ...