For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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3
votes
2answers
72 views

Prove $A = (A \setminus B) \cup (A \cap B)$

Prove $A = (A \setminus B) \cup (A \cap B)$ Logically, this is clearly true. I can explain why: start with $A$, remove all elements in $B$ and then add in any elements in both $A$ and $B$, which ...
1
vote
1answer
18 views

Proving increasing function, base < 1, exponent increasing

For a fair lottery game where the odds of $1$ ticket winning are $1$ in $p$, where you can spend a total of $K$ dollars, and where you will spread your ticket purchases equally among $n$ draws, prove ...
1
vote
2answers
20 views

Solution verfication and two small cardinality questions

I'm studying to my final exam due to tomorrow, and I encountered several small problems. Determine the cardinality of the following sets: 1). $A$ is the set of all injective functions from ...
0
votes
0answers
40 views

How do we prove that, if $\mathcal{P}(A) \sim \mathcal{P}(B)$, then $A \sim B$? [duplicate]

The converse--if $\ A \sim B$ then $ \mathcal{P}(A) \sim \mathcal{P}(B)$--is very easy to prove. I can't see an immediate, simple proof for the converse case. It seems like a potentially good strategy ...
4
votes
0answers
97 views

Saddle Points on Matrices

Let $n$, $m$ be positive integers. Suppose that $A$ is a $2$ x $n$ or an $m$ x $2$ matrix and that it has a saddle point. Show that among the saddle points of $A$ there exists at least one which ...
5
votes
5answers
783 views

Proof of infinitely many primes, clarification

Proof: The proof is by contradiction. Suppose there are only finitely many primes. Let the complete list be $p_1,p_2,\dots,p_n$. Let $N = p_1p_2 \dots p_n+1$. According to the Fundamental Theorem of ...
0
votes
0answers
30 views

Matrix Saddle Points and Dominance

I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it can be used in games. I wanted to know how to prove a problem like ...
0
votes
1answer
33 views

Proof regarding division with a remainder

Let $a\in\mathbb{Z},n\in\mathbb{N}$. If $a$ has a remainder $r$ when divided by $n$, then $a\equiv r\pmod n$ I've done some of these questions before with modulus and division, but I'm unsure of how ...
2
votes
5answers
69 views

Proof regarding divisibility

if $n\in\mathbb{Z}$, then $4$ does not divide $(n^2 - 3)$ I'm not sure how to approach this question, I know how to do questions that involve proving that it does divide but I'm unsure of how to do ...
0
votes
2answers
24 views

Modulus related proof help

I need to prove this via either direct proof, or contrapositive. Unsure of the best way to approach this. if $a \equiv b\mod n$ and $c \equiv d\mod n$, then $ac \equiv bd\mod n$ So far I have: ...
1
vote
3answers
53 views

Proof for modulus via direct or contrapositive

I have to prove the following via direct proof or via contra positive. For $a,b\in \mathbb{Z} $ it follows that $ (a+b)^3 \equiv a^3 + b^3 \mod 3$ I'm unsure of the best way to approach this ...
0
votes
3answers
64 views

Convergence of sequence method, Math behind intuition

Now I want to find convergence of a sequence: $$ \lim_{n \to \infty} \sqrt[n]{4^n + 5^n}$$ Now I am pretty sure I have solved this using logic on inspection: $4^n \ll 5^n$ as $n\rightarrow\infty$, ...
0
votes
4answers
68 views

Show that * is associative

Could you show me how to prove the following to be associative? Please take me through the process step by step. $$a*b=a+b+2ab$$ Where $*$ is a binary operation and $a$ and $b$ are real numbers. I ...
2
votes
2answers
115 views

Proof of Sylow's theorem.

I read this proof of Sylow's theorem in Rotman's "An introduction to the Theory of Groups" and I don't understand what is the argument in the second paragraph (the one in the green box) for. Isn't ...
-1
votes
1answer
58 views

Mean and Standard Deviation self thought problem

I am 13 years old trying to teach myself about standard deviation and was wondering how this problem would look like. I know I am young to be learning this but I was reading about this and got ...
1
vote
0answers
15 views

Existence of a particular transformation

I've a set of data points $S = \{ x | x\in [0,1]\}$ (i.e. real values from the unit interval). In some cases I've big clusters in the data and I want to spread the values in between the unit interval ...
0
votes
3answers
186 views

Proof of $n^{th}$ derivative Test

Proof needn't be a rigourous , but should give an insight of how $n^{th}$ derivative test (higher order derivative test) works as i know how to use it in application but i don't much understand it ...
2
votes
4answers
88 views

Easier Proof of $\sin{3\theta} + \sin\theta = 2\sin{2\theta}\cos\theta$

I am curious to see whether anybody can give me a proof that takes less steps. Here is how I did it: $$\sin{3\theta} + \sin\theta = 2\sin{2\theta}\cos\theta$$ LHS $$\eqalign{\sin(2\theta + \theta) ...
5
votes
2answers
90 views

Explicit expression for homeomorphism and homotopy equivalence

In many cases of topology when one needs to show two spaces $X$ and $Y$ are homeomorphic or homotopy equivalent, one uses some description instead of constructing an explicit homeomorphism or homotopy ...
2
votes
1answer
39 views

Set difference between power sets

Show that if $A$ and $B$ are sets then $\varnothing \notin \mathcal P(A) − \mathcal P(B)$ . My Attempt: If $A$ and $B$ are sets, then $ \varnothing \subseteq A $ since the empty set is subset of ...
3
votes
5answers
314 views

Prove $\sin(-x) = -\sin(x)$

I'm looking for a really basic proof of $\sin(-x) = -\sin(x)$. The proof should pretty much only employ basic trigonometry. Thanks
0
votes
1answer
22 views

Geometric proof of dot product distributive property

I'm working my way through a text book for fun in order to keep my math brain fresh and came across this simple yet perplexing problem. "Demonstrate geometrically that the dot product is ...
1
vote
1answer
39 views

Induced bijections of combinatorial species

I'm doing this exercise: Prove that $\mathcal{S}[\beta]$ is a bijection. Here, $\beta:M\rightarrow N$ is a bijection of finite sets, $\mathcal{S}$ is a species, and ...
7
votes
4answers
587 views

What is the correct way of disproving a mathematical statement?

This question is motivated by my midterm exam. In this exam there was a question as follow: Question: If the following statement is true, prove it, otherwise disprove it. If $\mathbf{u}$ and ...
0
votes
1answer
46 views

Proof of AM-GM Inequality with lemmas

I need to prove the AM-GM Inequality using a few specific lemmas that I have already proven. I'm mostly just unsure what to do next and how to tie it all together at the end to finish the proof. Here ...
1
vote
1answer
48 views

How can I know which theorem to use to prove another one?

In class this year a part of what we do is re learning theorems etc from previous years, but a more rigorous way. However, when I suggest a way to prove those theorems/properties/..., I often get an ...
1
vote
1answer
39 views

prove $(\Bbb{R},\{X\mid X\subseteq \Bbb{R} \wedge \forall y\in X(\exists r,s \in \Bbb{R}( y\in ]r,s[\wedge ]r,s[\subseteq X))\})$ is topological space

I must prove the following: Prop.: $(\Bbb{R},K:=\{X\mid X \subseteq \Bbb{R} \wedge \forall y \in X(\exists r,s \in \Bbb{R}( y \in ]r,s[ \wedge ]r,s[ \subseteq X))\})$ is topological space Proof.: ...
2
votes
1answer
53 views

Reverse Hex board game winning strategy

I just wanted to know the winning strategy to this question: In a reverse Hex board game I know it means where the player who first forms a path between his/her edges loses. Find a winning ...
6
votes
4answers
656 views

The role of 'arbitrary' in proofs

Generally, when one is going to prove a result regarding a set of elements, they begin their proof with those first few pleasing words: "Suppose...is an arbitrary element in..." My question is, why ...
1
vote
0answers
40 views

What is the relationship between division and proving an integer is odd?

I am trying to use proof by contradiction to prove: $101$ is an odd integer. I know that the first step is to assume that $101$ is even, so: $101 = 2q, q \in \mathbb{Z}$ Then I am stuck. I don't ...
1
vote
3answers
93 views

Prove: $\sum_{x=0}^{n} (-1)^x {n \choose x} = 0$

Is there a quick, fancy, way of proving sums such as this? Prove that: $$\sum_{x=0}^{n} (-1)^x {n \choose x} = 0$$ A recent homework assignment I turned in had a couple problems similar to the ...
2
votes
2answers
100 views

How to prove that the law of the excluded middle is necessary?

This is a follow up question to this answer by Carl Mummert to the question whether every proof with contradiction can also be proved without contradiction. As Carl Mummert pointed out, there are ...
4
votes
6answers
551 views

Intro to Real Analysis

I am having trouble proving the following: if $a < b$, then $a < {a+b\over2} < b$. I started with the Trichotomy Property and getting to where $a^2>0$, but then I do not know where ...
1
vote
1answer
23 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
0
votes
0answers
28 views

Issue with the geometric proof of lim_{x -> 0} sinx/x = 1

When proving $\displaystyle\lim_{\theta \to 0} \frac{\sin\theta}{\theta} =1$, I have been taught to use a sector with radius 1. How rigorous is this proof if we have not considered a radius of any ...
1
vote
1answer
36 views

Endomorphism with rank $r$ annihilates degree $r+1$ polynomial

Let $f$ be a linear transformation of $\mathbb R^n\to \mathbb R^n$ that has rank $r$. Prove the existence of a degree $r+1$ polynomial that annihilates $f$ I have a proof : consider $g$ ...
1
vote
3answers
23 views

Proving GCD property

The questions says Let a and b be integers. Show that if (a,b) = 1 then (a,a + b) = 1. I'm stuck on how to do this proof. I tried using the fact that ax+by=1 but that didn't seem to help.
2
votes
2answers
51 views

Question about the density of Q in R

So I was looking over a density that shows that the rational numbers are dense in the real numbers. If $0< a <b$, with with $a,b$ real numbers, then I understood why we can chose n such that: ...
0
votes
0answers
42 views

Proof by Induction for Fundamental Thm of Arithmetic

Use induction to make our proof of the Fundamental Theorem of Arithmetic more rigorous. Recall that $p$ is prime iff for all $a,b\in\mathbb Z:p\mid(ab)$ implies $p\mid a$ or $p\mid b$. Prove that ...
1
vote
3answers
30 views

Verification of Proof strategy

I am tasked with proving the following : $$A \cap B^c \subseteq (A \cap B)^c$$ I came up with the idea of using a combination of De Morgan's laws, rule simplification and rule of addition to prove ...
0
votes
1answer
20 views

Minimum k-spanning tree including a given node

Given a Graph (V, E), it is very easy to find the minimum spanning tree using Kruskal's Algorithm. A k-minimum spanning tree is restricted to k nodes, and finding it is actually NP-hard. However, ...
3
votes
0answers
36 views

Prove Differentiation Multivariable

Given $f(x,y) = \frac{ xy^2}{x^2 +y^2}$ From defintion we know it is differentiable if: $\lim_{h\to 0}\frac{F(X+h)-F(X)-c*h}{|h|}$ exists, where $c$ is the gradient of the function. I have ...
0
votes
1answer
52 views

Winning or Non-losing strategy for A or B

Find a winning or a non-losing strategy for the following game: Consider $25$ sticks arranged in a $5$ x $5$ square. Players alternately take any number of sticks from a single row or column. At ...
2
votes
1answer
32 views

$\frac{|| \overline{AM}||}{|| \overline{AB}||}=\frac{|| \overline{AN}||}{|| \overline{AC}||}=\frac{|| \overline{MN}||}{|| \overline{BC}||}$

$\Delta ABC$ is a triangle, $M$ is a point in the segment $\overrightarrow{AB}$ and $N$ is a point in the segment $\overrightarrow{AC}$, such that $\overrightarrow{MN}$ is parallel to ...
1
vote
1answer
119 views

Union of convex sets is convex: Strategy?

I am working in convex geometry for the summer with little experience beforehand. It's a lot of fun but it does mean I don't know some of the basic things. I know that it is not generally true that ...
1
vote
3answers
62 views

Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$ [duplicate]

Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$ How would you do the inductive step for this proof? I have the base case done.
1
vote
5answers
48 views

Matrix Power Formula

Prove that for a fixed $a \in \mathbb{R}$ we have the matrix power formula for all $n \in \mathbb{Z}_+$: $$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^n = \begin{pmatrix}a^n & ...
1
vote
2answers
56 views

Prove that n!+1 contains a prime factor greater than n and use this to prove that there are infinte many primes [duplicate]

Prove that $n!+1$ contains a prime factor greater than $n$ and use this to prove that there are infinitely many primes. I said assume that $n!+1$ contains a prime $p$ which is less than or equal to ...
0
votes
4answers
147 views

Singleton sets are a subset

I am tasked with prove the following elementary result. I am concerned about being rigours enough in my proof: $$a\in S \iff \{a\} \subseteq S $$ My Attempt: Suppose $\{a\} \subseteq S $ . Then ...
1
vote
2answers
87 views

Hex game winning strategy

I was teaching myself how to play a hex board game by reading some books a couple days ago. I learned how to do $2$ x $2$ and $3$ x $3$ hex games by starting at the principal diagonal. I wanted to ...