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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In every set of $14$ integers there are two that their difference is divisible by $13$

Prove that in every set of $14$ integers there are two that their difference is divisible by $13$ The proof goes like this, there are $13$ remainders by dividing by $13$, there are $14$ numbers ...
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2answers
99 views

Understanding why $a+b\sqrt {2}\neq \sqrt {3} $

I want to intuitively understand why $a+b\sqrt {2}\neq \sqrt {3} $ for $a, b \in \mathbb Q $ I really have no intuition regarding this matter, and have to deal with similar concepts regularly while ...
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1answer
40 views

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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1answer
54 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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4answers
65 views

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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23 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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3answers
151 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
50 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
54 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
26 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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0answers
48 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 ...
2
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3answers
50 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
73 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 ...
2
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2answers
72 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
44 views

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
178 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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0answers
14 views

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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0answers
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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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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34 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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6answers
9k views

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
27 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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0answers
22 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
47 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
45 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
234 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 ...
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2answers
411 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 ...
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1answer
75 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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0answers
42 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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35 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 ...
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1answer
63 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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3answers
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
42 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 ...
3
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2answers
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
67 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 ...