Questions tagged [intuition]
Mathematical intuition is the instinctive impression regarding mathematical ideas which originate naturally without regard to formal mathematical proofs. It may or may not stem from a cognitive rational process.
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Theorem : For any two real numbers $a,b$ with $a < b: [a,b]$ is uncountable
This is a Theorem in the book, "A Course on Borel Sets" [[ SM Srivastava 2001 ]] I was reading and I understood nearly everything but a statement which gives the contradiction, I am ...
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An Apparent Paradox on Cardinalities of Infinite Sets
I was going through this link and a comment by user fkraiem mentioned that a characteristic property of an infinite set is that it has the same cardinality as at least one of its proper subsets. But ...
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Difference between the elements of $\mathbb C$ and $\mathbb R^2$ [duplicate]
I know that, as sets, $\mathbb C$ and $\mathbb R^2$ are exactly the same set, and that the difference is about the structure: $\mathbb C$ is $\mathbb R^2$ with a field structure.
Does this mean that, ...
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Motivation behind the definition of random variable
I'm currently reading A First Look at Rigorous Probability Theory by J. S. Rosenthal, and I just got to the definition of random variable:
Definition 3.1.1 Given a probability triple $(\Omega,\...
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Nice way to show that the rationals are not complete
I have been thinking about how to show that the set $ S = \{ \text{x is rational} , x^2 < 2 \} $ has no least upper bound in the rationals space, in a way that is intuitive for me. In that sense, I ...
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What does it mean for a predicate to be ground?
I'm reading a paper on the AI tech of HTN planning. In there, the authors state:
A predicate, which evaluates to true or false, consists of a predicate symbol $p \in P$, where $P$ is a finite set of ...
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Is it better to hire $1$ fast barista vs $2$ slower baristas?
This is another math puzzle I heard today.
Consider a M/M/K queue (https://en.wikipedia.org/wiki/M/M/c_queue) in a cafe. Lets say the cafe has a rule that each queue is FIFO (first in first out), each ...
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Relation about line picking on $n$-sphere and $n$-ball
Define $ S[n] $ as the expected distance between two points uniformly distributed on the surface of a $n$-sphere of radius $1$ (here you can find the explicit values). Now define $ B[n] $ as the ...
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Meaning of permutation order in QR decomposition with pivoting
I used Matlab's qr function to calculate the qr decomposition of $A^H$, using:
[Q, R, P] = qr(A', 'vector').
This method uses ...
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How did Euler know $X=6^{128} + 1$ is divisible by $Y=257$
I am working through a worksheet talking about some of Euler's results and how $X = a^{2^n}+1$ is not necessarily prime.
One example that Euler gave was that $X = 6^{128}+1$ is not prime since it is ...
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How to transform a triangle into a circle?
I am trying to come up with a function that allows me to "transform a triangle into a circle".
For example, given a circumference with radius $1$ and center $(0,0)$ , imagine a triangle ...
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How do Algebraic functions happen to be important in the study of Riemann surfaces and arise naturally?
I am a Research Scholar and I am trying to explore the connection between Algebraic functions and Branched coverings in the study of Riemann surface.Since,I am a beginner,I would like to have some ...
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If $|A| \le |B| \le |C|$ and $A \subseteq C$ does there exist $D$ with $A \subseteq D \subseteq C$ and $|B| = |D|$?
Suppose that $|A| \le |B| \le |C|$ and $A \subseteq C$. Assuming ZF axioms, but not AC, does there necessarily exist a set $D$ such that $A \subseteq D \subseteq C$ and $|B| = |D|$?
Here is my proof ...
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Let $X$ be a set with $n$ elements. Prove that $ \sum_{Y, Z \subseteq X}|Y \cap Z|=n \cdot 4^{n-1} $
Let $X$ be a set with $n$ elements. Prove that $$
\sum_{Y, Z \subseteq X}|Y \cap Z|=n \cdot 4^{n-1}
$$
The sum is over all possible pairs $(Y, Z)$ of subsets of $X$.
I do not need a solution, but ...
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What is the map of the Stone-Čech compactification of the rationals to that of the reals like?
This is a kind of followup question to this old one.
Let $\mathbb{Q}$ and $\mathbb{R}$ have their usual (Euclidean) topology, and let $\beta\mathbb{Q}$ and $\beta\mathbb{R}$ stand for their respective ...
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What is a "zero term" of a polynomial?
On the Wikipedia page for homogeneous polynomials, it is stated that "a homogeneous polynomial [...] is a polynomial whose nonzero terms all have the same degree."
What is a "zero term&...
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Intuition behind degree of minimal polynomial
I'm learning linear algebra from Linear Algebra Done Right. The book gives a proof that the degree of the minimal polynomial of an operator on V is at most the dimension of V, but the proof feels ...
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Smallest natural number unrepresentable by fifty letters
What is the problem of the proof of this claim? I also think it is really strange, but I can't find the problem.
Claim:
All natural numbers can be uniquely represented using fifty letters.
Proof:
...
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Dummit Foote, Group Theory, Section 1.6, Problem 23 [closed]
Dummit Foote, Group Theory, Section 1.6, Problem 23.
Let $G$ be a finite group which possesses an automorphism $\phi$ such that $\phi(g) = g$ if and only if $g = 1 $. If $\phi^2$ is the identity map ...
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Convex sets - unable to prove that the two properties are the same.
Convex sets have property that a line between any two points, lies in the set.
This is stated algebraically here as:
Given an affine space E, ... a subset V of E is
convex if for any two points $a, b ∈...
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Intuition for determinant over other fields/rings [duplicate]
Over $\mathbb R^n$, the standard intuition given to the determinant is that it measures the signed area of the image of an unit cube. But determinants can be more generally defined for endomorphisms ...
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Why aren't all angles in (n-)spherical coordinates defined relative to the x-axis?
2D polar coordinates are generally measured as $(r, \theta)$ where $\theta$ is the angle measured counter-clockwise from the x-axis, and ranges from $0$ to $2\pi$.
Then, spherical coordinates extend ...
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Connection between Hilbert function and Euler characteristic
I strongly suspect that there is a connection between the Hilbert function $H_M(d)$ for a graded $K[x_0,...,x_n]$-module $M$ and the Euler characteristic of a topological space. For the obvious ...
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Proof of a definite integral whose result is the difference of arithmetic and geometric means
This might have been already asked in this site but I can't find it. So here's the integral:
$$\int_{r_\text{min}}^{r_\text{max}} \sqrt{\left(1-\frac{r_\text{min}}{r}\right)\left(\frac{r_\text{max}}{r}...
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Two different answers for $ I=\int \frac{dx}{1+2\sin^2x+3\cos^2x} $
Evaluate :
$$ I=\int \frac{dx}{1+2\sin^2x+3\cos^2x} $$
My attempt :
By multiplying Numerator and Denominator by $\sec^2x$ and solving I get the value of the integral as $$I_1=\frac{1}{2\sqrt3} \tan^{-...
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