Mathematics

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1500 questions
73
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4 answers

Can there be two distinct, continuous functions that are equal at all rationals?

Akhil showed that the Cardinality of set of real continuous functions is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree everywhere, since the rationals are dense in the…
Charles Stewart
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5 answers

What does proving the Riemann Hypothesis accomplish?

I've recently been reading about the Millennium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even fully grasp the problem, but seeing the hypothesis and the other problems I wonder: what practical use will a…
Mythio
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73
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2 answers

Is Serge Lang's Algebra still worth reading?

Is Serge Lang's famous book Algebra nowadays still worth reading, or are there other, more modern books which are better suited for an overview over all areas of algebra? EDIT: My main concern is that the first edition of Algebra is already 48 years…
Dominik
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Where does the word "torsion" in algebra come from?

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry (which relates back to the ordinary use of the…
Dan Piponi
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16 answers

Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?

I don't know if my textbook is written poorly or I'm dumb. But I can't bring myself to understand the following definition. A real number is a cut , which parts the rational numbers into two classes. Let $\mathbb{R}$ be the set of cuts. A cut is a…
God bless
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3 answers

Recognizable vs Decidable

What is difference between "recognizable" and "decidable" in context of Turing machines?
metdos
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3 answers

First-Order Logic vs. Second-Order Logic

Wikipedia describes the first-order vs. second-order logic as follows: First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables…
Sadeq Dousti
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73
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5 answers

In combinatorics, how can one verify that one has counted correctly?

This is a soft question, but I've tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly. It's easiest to explain with an example, so I'll give…
Stephen
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73
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33 answers

How to explain to my daughter that $\frac {2}{3}$ is greater than $\frac {3}{5}$?

I was really upset while trying to explain to my $8$ -year-old daughter that $\dfrac 23$ is greater than $\dfrac 35.$ She doesn't know decimals, so she can't realize that $\frac 23 = 0.66$ and $\frac 35 = 0.6.$ She claims that since $3$ is greater…
Akam
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73
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7 answers

The relation between trace and determinant of a matrix

Let $M$ be a symmetric $n \times n$ matrix. Is there any equality or inequality that relates the trace and determinant of $M$?
TPArrow
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9 answers

What is the most expensive item I could buy with £50?

I was set the following question during the discrete mathematics module of my degree and despite my instructor explaining his working to me I still disagree with the answer he says is correct. Can someone please help me either understand where my…
Sam
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10 answers

Why is the tensor product important when we already have direct and semidirect products?

Can anyone explain me as to why Tensor Products are important, and what makes Mathematician's to define them in such a manner. We already have Direct Product , Semi-direct products, so after all why do we need Tensor Product? The Definition of…
anonymous
73
votes
8 answers

Is linear algebra laying the foundation for something important?

I'm majoring in mathematics and currently enrolled in Linear Algebra. It's very different, but I like it (I think). My question is this: What doors does this course open? (I saw a post about Linear Algebra being the foundation for Applied…
Mallory
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73
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14 answers

Are proofs by contradiction really logical?

Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction. My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"? That, to me, seems the…
Simp
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2 answers

Continuous functions do not necessarily map closed sets to closed sets

I found this comment in my lecture notes, and it struck me because up until now I simply assumed that continuous functions map closed sets to closed sets. What are some insightful examples of continuous functions that map closed sets to non-closed…
Aaa
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