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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30
votes
5answers
3k views

Is it okay to reverse engineer proofs in homework questions?

In a linear algebra text book, one homework question I received was: Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$. Where $\mathbf{a}$ and ...
1
vote
1answer
30 views

Arithmetic progression and proofs

Here is the question I am stuck on An arithmetic progression of integers an is one in which $a_n = a_0 + nd$, where a_0 and d are integers and n takes successive values 0, 1, 2.... Proof that if one ...
0
votes
1answer
33 views

Proof by induction and inequalities

I am stuck on this question: given $a_1a_2≤(\frac{a_1+a_2}{2})^2$ prove by induction of m that $$a_1a_2...a_p≤(\frac{a_1+a_2+...+a_p}{p})^p$$ where $a_i$ are all positive and real and $p=2^m$ (an ...
1
vote
1answer
24 views

Proof of isometries and inverses on the plane

I am taking a course on Intuitive Geometry. I am quite new to intuitive proofs however feel I've done pretty well thus far. Here is my theorem: Prove: That every isometry has an inverse. $Proof.$ ...
1
vote
1answer
59 views

Proof by contradiction: logarithm

I need to prove by contradiction that $\log_2(3)$ is irrational. I'm really unfamiliar with logs to be honest, it's been awhile since I've done them and I'm unsure of how to approach this. Any help ...
0
votes
3answers
41 views

Proof by contradiction proving both numbers are not odd.

I have to do a proof by contradiction: Suppose $a,b,\in\mathbb{Z}$. If $4| (a^2 + b^2)$ then a and b are not both odd. So far I know that I need to prove that if $4|(a^2+b^2)$ then a and b are both ...
0
votes
2answers
61 views

Proof by contradiction for a set question

I have a statement I need to prove by contradiction: If A and B are sets then A intersect (B-A) = {} (empty set). None of the questions I've ever done for this class are like this so im not really ...
0
votes
0answers
18 views

Proof Strategy for Proving an Inequality Involving Products

I'm working on a proof in my complex analysis course that involves showing that $$ A \cdot B \leq C \cdot D $$ ($A$, $B$, $C$, and $D$ are expressions involving the moduli of complex numbers). My ...
1
vote
1answer
18 views

meaning of some general words in math theorems

I want to prove a theorem which starts with the statement "Let ... then there exists ... ." Now I want to know the meaning of let in this kind of theorems. "Let" means for every or for some of ? How ...
0
votes
1answer
182 views

If $X' \leq X$ almost surely, is it possible to prove that $P(X = s) \geq P(X' = s)$?

With respect to my previous question, let us define $X$ as: $$ X = \sum_j^r l^j Y^j, $$ where $l^j \geq 0$ and $Y^j$, $j = 1, \ldots, r$ is a Bernoulli random variable which takes on values in ...
2
votes
1answer
26 views

Group divisibility question

I have the following question which I can't make sense of, here is the entire question: If $G$ is a group, $b\in G, o(b)=k$ and $b^n = e$, show that $k|n$ What is $o(b)$? Please help.
3
votes
2answers
35 views

Basis and dim of the set of all $n\times n$ symmetric matrices.

An $n \times n$ square matrix $A$ is called symmetric if $A^T = A$ Show that the set of all $n \times n$ symmetric matrices, denoted $S$, is a subspace of $M_n(\mathbb{R})$. Give a basis for $S$ ...
1
vote
1answer
38 views

Proof of coset and normal subgroup

I have this question: Let $G$ be a group, $a,b\in G$ and let $H$ be a subgroup of $G$. i) Give the definition of the coset $aH$ ii) Prove that $aH = bH$ if and only if $a^{-1}b\in H$ ...
0
votes
2answers
47 views

If $g \circ f$ is injective, so is $g$

If $g \circ f$ is injective, so is $g$ I don't think this is true. I think that $f$ has to be surjective. So I am going to try to prove that: If $g \circ f$ is injective, and $f$ is ...
0
votes
2answers
84 views

Prove that a function is bijective

So, the problem sounds like this. You have two bijective functions $f:\mathbb{N} \to A$, $g:\mathbb{N} \to B$. We define the function $ h:\mathbb{N} \to A \cup B $, defined as: $$ h(n) = ...
0
votes
1answer
12 views

A question about the proof of the limit comparison test for series

A question about the proof of the limit comparison test for series: http://en.wikipedia.org/wiki/Limit_comparison_test About the the last part: $b_n(c-\epsilon)<a_n<(c+\epsilon)b_n$, to ...
19
votes
2answers
307 views

How to prove $ \lim_{n \to \infty} e^n \cdot \left( \sum_{k=0}^{n-1} ({k-n \over e})^k/k! \right)- 2 \cdot n = \frac 23$?

I observed for the function $$ f(n)= e^n \sum_{k=0}^{n-1}\left(\dfrac{k - n}{e}\right)^k \cdot \dfrac{1}{k!} \tag 1$$ with small $n$ that ...
5
votes
5answers
158 views

Show that $\frac{\sqrt{8-4\sqrt3}}{\sqrt[3]{12\sqrt3-20}} =2^\frac{1}{6}$

This was the result of evaluating an integral by two different methods. The RHS was obtained by making a substitution, the LHS was obtained using trigonometric identity's and partial fractions. Now I ...
3
votes
2answers
67 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
17 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
19 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
95 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
772 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
29 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
67 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
49 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
60 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
65 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
111 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
53 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
183 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
82 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
85 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
35 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
301 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
20 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
37 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
580 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
45 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
45 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
38 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
51 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
653 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
96 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 ...