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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Can an element of a power set $2^A$ be a subset of $2^A$?

This question is continued from a previous thread I started, but it had more than one question so I had to move the other question here. For this example consider an injective map $f: A \to 2^A$ then ...
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50 views

If $A$ is a non-empty set and $2^A$ is the power set of $A$. Is $2^A \subseteq A$?

I'm aware that if there exists an injective map $f: A \to 2^A$ then for each element $a\in A$ $\exists$ $f(a)\subseteq A$. But does this also mean $f(a)\subseteq 2^A$? I ask this because when ...
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Why Mendelson axiom schemas are true?

I'm taking course in logic. The book is available here I don't understand why is Mendelson axiom schemas are the way they are. For example implication creation schema $φ ⇒ (ψ ⇒ φ)$ My thoughts ...
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22 views

mean value theorem applications

it is said that the mean value theorem prove that the graph between $[a,b]$ has a point where it is equal to the average change of the graph, (sorry for not being accurate) is there more information ...
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138 views

What do groups and rings “look like”?

Taking undergraduate physics courses, I had to deal with Euclidean vectors often. In classes like Calc III, the concept was also there. I'm not sure if this is why, but I've always had a more ...
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1answer
45 views

Why Does The Taylor Remainder Formula Work?

I've been studying calculus on my own and have come across Taylor series. It is very intuitive until I came across the remainder part of the formula where things got fuzzy. I understand why the ...
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1answer
53 views

Why are power sets called power sets?

Why are power sets called power sets? What is so powerful about them?
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1answer
62 views

Trouble understanding what a measure-zero set is.

To begin with some context, I haven't had any exposure to measure theory yet. I solved the following problem. A set $A\subset \mathbb R$ such that $\forall \epsilon >0$, there exists countably ...
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1answer
20 views

Justification, intuition and motivation for cardinal arithmetic [closed]

What justifies, provides intuition and motivation for weird cardinal arithmetic?
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2answers
30 views

What is the best way to explain setting a restriction on $\delta$ in $\epsilon$-$\delta$ proofs?

I'm trying to prepare a somewhat informal lesson striving to provide an intuitive understanding of why for some limit proofs, we have to set an upper bound on $\delta$. For example, here's part of ...
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1answer
25 views

Geometric proof for $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$

Is there an geometric proof for the following identity? $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$. The norm here is normal Euclidean norm, and $u,v$ are vectors.
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46 views

Physical analogies of a math concepts [closed]

In a post Terence Tao explained a very nice way to think about convolution and noted that "one should try to use physical intuition to model mathematical concepts whenever one can". I found this very ...
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3answers
49 views

Why is $\mathbb T\cup\mathbb A = \mathbb Q \cup \mathbb I =\mathbb R$?

Where $\mathbb T $ is the set of transcendental numbers, and $\mathbb I $ is the set of irrational numbers and $\mathbb A $ is the set of algebraic numbers. The sets $\mathbb Q$ and $\mathbb R$ have ...
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2answers
72 views

Ordinals - motivation and rigor at the same time

Can someone provide a description of ordinals within ZFC in a rigorous way that exhibits motivation? Every description or explanation I see in the literature or on the Internet is either too formal ...
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3answers
156 views

Understanding the use of the Cartesian Product in the proof of $|\mathbb R\times \mathbb R|=|\mathbb R|$

Where the Cartesian Product of two sets $\mathbb A$ and $\mathbb B$ is such that $\mathbb A\times \mathbb B=\{{ (a,b)|a \in \mathbb{A}, b \in \mathbb{B}\}}$ In trying to understand the proof that ...
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1answer
45 views

matrices with determinant equals to one

we already know what does it mean the determiant of a matrix is null, it's not invertible ! but what about matrices with determinant equals to $1$ ?! I know that the determinant of matrix is the ...
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1answer
54 views

What is the practical meaning of derivatives? [closed]

I mean practically integration means sum of all components, and the integral can be visualized as the area below a curve. Is there a similar intuition or geometric meaning of the derivative?
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6answers
115 views

Intuitively understanding $\sum_{i=1}^ni={n+1\choose2}$

It's straightforward to show that $$\sum_{i=1}^ni=\frac{n(n+1)}{2}={n+1\choose2}$$ but intuitively, this is hard to grasp. Should I understand this to be coincidence? Why does the sum of the first ...
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2answers
69 views

Proofs of theorems, where picture is sufficient

A while ago I have had the pleasure to come across those lectures of Topology & Geometry by Dr Tadashi Tokieda (I do recommend watching at least the first lecture, both parts). My question is ...
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1answer
32 views

Geometric intuition of the mean value theorem of several variables

Mean value theorem Let $f:U\to \mathbb R$ be defined in the open set $U\subset \mathbb R^n$. Suppose the segment $[a,a+v]$ be contained in $U$ and the restriction $f|_{[a,a+v]}$ be continous ...
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5answers
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Where does the constant increase by 2 of differences between integer square values come from?

$1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$, etc... Looking at the difference between those square values, we get: 3, 5, 7, 9, etc... The difference from one (integer) square to the ...
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2answers
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Improve/extend my attempted intuitive explanation for why terms in determinant calculations have alternating signs

The determinant of a shape defined by points $(a,b)$ and $(c,d)$ as labelled in the gif below is $\left|\begin{matrix}a&c\\b&d\end{matrix}\right| = ad-bc$ The following process is the ...
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1answer
60 views

Why are the Cauchy-Riemann equations in polar form 'obvious'?

In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'. ...
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6answers
175 views

Intuition: “If P then Q” = 'Not P or Q' [closed]

I already understand, and so ask NOT about, the Conditional Law: $P \Rightarrow Q \; \equiv \;\lnot P \vee Q$. But what's the intuition? Because I ask only for intuition, please do NOT prove formally ...
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Linear functionals defines projection operators?

The way I always understood linear functionals on a vector space $V$ is to consider then as measuring objects which give projections when they are given vectors. Now I wanted to make this a little bit ...
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5answers
80 views

Intuition: Why is the biconditional true if both statements are false?

I already know that a false statement implies anything. Because I ask only for intuition, please do NOT prove this or use truth tables (which I already understand). Source: p 333, A Concise ...
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58 views

How to think/see point-set topology abstractly?

I've started learning point-set topology this semester. I've learned basic material about: topology on a set topological space open sets closed sets clopen sets closure neighborhoods interior point ...
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3answers
78 views

When is it sufficient to use logic as proof for an intuitive answer

Say I have the following limit $$\lim_{x\rightarrow\infty}\frac{3^x}{e^{x-1}}$$ In this case it's simple enough to write it as $\lim_{x\rightarrow\infty}{3^x}{e^{1-x}}$ and then show it approaches ...
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45 views

How to intuit 'unless'? [duplicate]

Foreword: The following (seeking only intuition) does NOT duplicate this (which explains with formal proofs.) I already know, and so ask NOT about, the proof of: $A$ unless $B$ = $A$ if not $B$ = ...
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0answers
33 views

Intuitive probability theory!?

Recently I saw something where someone had a paper with several lines on it and a needle. The length of the needle was the same as the distance between the lines. They then proceeded to say that when ...
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1answer
48 views

Why is a cartesian morphism called cartesian?

I am reading about fibred categories. After reading the definition of "vertical" morphism, I can imagine why they are named like that. What about "cartesian" morphisms? What is cartesian about them? I ...
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What did Alan Turing mean when he said he didn't fully understand dy/dx?

Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult ...
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1answer
25 views

Barendregt's Substitution Lemma (lambda calculus)

I am struggling to put words on an idea used in Barendregt's Substitution Lemma's proof. (available here) The lemma states that: If x≠y and x not free in L and M, L are $\lambda$-terms: then ...
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1answer
33 views

Interpretation of alternative group structures on a given group

Let $(G,e,\circ) $ be a group with $e$ the identity element and $a \in G$ and $\circ$ the group operation. Then we can form a new group $(G_a,a,\circ_a)$ with the same underlying set as $G$ and $x ...
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21 views

Antiderivative of a position function.

Let's say you had some sort of position function, i.e. a moving ball, $f(x)=-2x^{2}+3x-7$. What is the intuitive (relates to the situation) definition of the antiderivative of this function? It ...
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2answers
45 views

Index in complex analysis is an integer : Intuition?

In complex analysis course, we prove that given a closed path $\gamma$ and $a\notin \gamma^*$ the following number: $$ \frac{1}{2i\pi}\int_{\gamma}\frac{dz}{z-a} $$ is an integer. The integral can ...
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0answers
44 views

How to show $k$-algebras are isomorphic in practice

I am working through some problems which require me to show when some $k$-algebra ($k$ a field) maps are isomorphisms. Unfortunately, I've got myself a bit confused with definitions and the like, and ...
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4answers
233 views

Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
6
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2answers
410 views

Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse? Help would be ...
8
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1answer
71 views

What the curvature $2$-form really represents?

Let $(E,\pi,B)$ be a principal bundle with structure group $G$. The connection $1$-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as ...
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41 views

connection among big-M, Lagrangian, Pentalty Method, and Augmented Lagrangian

In the context of solving linear programs, the big-M method refers to adding additional variables to the problem such that there is, as far as I understand it, a trivial basic feasible solution. In ...
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2answers
88 views

How to intuit 'only if'?

I already know, and so ask NOT about, the proof of:   $A$ only if $B$   =   $A \Longrightarrow B$. Because I ask only for intuition, please do NOT prove this or use truth tables. My problem: I try ...
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0answers
34 views

Motivation For Tensor Product of R-Modules

I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important ...
2
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1answer
62 views

pointwise convergence of a filter on $\mathbb{R}^\mathbb{R}$

In my topology lecture we have defined pointwise convergence for filters on function spaces, say $\mathbb{R}^\mathbb{R}$. A filter $\varphi$ on $\mathbb{R}^\mathbb{R}$ converges pointwise to ...
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68 views

What are some illustrative examples that demonstrate how $\succ$ can differ in behavior from $>$ and/or $\geq$?

I really, really want to understand the generalization of metric spaces known as continuity spaces. Unfortunately, I always get tripped up right at the beginning. The problem is that I have little or ...
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1answer
40 views

intuition about cubic splines vs quadratic splines (degree 3 vs degree 2).

my intuition about quadratic(degree 2) splines is that by the help of its three variables (in each sub-interval) you can make a piecewise differentiable function on the whole interval. in the process ...
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1answer
106 views

What is an example of a proof that explicitly relies on the law of excluded middle?

I was talking with a friend about logic and I realized she might be an intuitionist. I was looking online for a proof that explicitly uses the law of excluded middle to see if she would have an issue ...
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1answer
63 views

Intuition behind the connection one-form

When we define a connection on one principal bundle $\pi: P\to M$ with structure group $G$ we define it as an association of one subspace $H_pP\subset T_pP$ for each $p\in P$ such that $T_pP = H_pP ...
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3answers
68 views

Where did the idea of hermite interpolation came from?

I am given the Hermite interpolation formula directly in my text book without ANY explanations about how it was first made (obviously it was somehow constructed for the first time with some sort of ...
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0answers
43 views

How to prove that $2^\sqrt2$ is irrational [duplicate]

I know the result is intuitive, but is there a way to mathematically prove it. I tried taking logarithm or tried proving by contradiction, but I don't know how to proceed. Thanks.