For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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1answer
61 views

Decomposition of a function into positive and negative parts and its integrability

1)Is it true that any function can be decomposed as a difference of its positive and its negative part as $f=f^{+}-f^{-}$ or that function should belong to $\mathcal{L}^{1}(\mathbb{R})$. Also if that ...
1
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1answer
59 views

Use of prime symbol in proof writing

Questions: So I looked through my course notes and saw this proof. I understand the content, but I'm confused about the use of the prime symbol. If we say that there is some $j'$ such that it is a ...
7
votes
3answers
102 views

How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
3
votes
0answers
40 views

Prove that $ 1+ \cos A + \cos B + \cos C = 0$. [duplicate]

If $A+B+C=180^\circ$ and $\tan \left[\dfrac{A+B-C} 4 \right] \tan \left[ \dfrac{-A+B+C} 4\right] \tan\left[\dfrac{A-B+C} 4 \right] =1$ then prove that $ 1+ \cos A + \cos B + \cos C = 0$ I ...
1
vote
1answer
41 views

Help to prove the condition that a right half-open interval is not empty

The right half-open interval is defined as: $[a,b) = \{x \in \mathbb{R}|a \le x \lt b\}$ I need to prove: $[a,b) \ne \emptyset \iff a<b$ My attempt: For $\Rightarrow$: $$\begin{align} ...
0
votes
1answer
32 views

Proof Writing Help: $P_UT=TP_U \Leftrightarrow U$ and $U^{\perp}$ are $T$-Invariant

I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my ...
1
vote
1answer
53 views

Proof that P is an Orthogonal Projection

I'm studying linear algebra using Axler's text and am stuck on 6.17. The problem is: Prove that if $P \in \mathcal{L}(V)$ is such that $P^2 = P$ and every vector in $\operatorname{null} P$ is ...
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3answers
150 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
2
votes
3answers
171 views

Proof that $\int \frac{1}{x}$ is $\ln(x)$

When I was learning Calculus AB and Calculus II/III at my high school, I noticed that our textbooks never gave a full fundamental proof that $\int \frac{1}{x}$ is $\ln(x)$, and rather said that when ...
2
votes
2answers
68 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
31
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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 ...
5
votes
1answer
111 views

How to prove the inequality?

Given $0<x<1$, $0<a<b<1$, and $a+b<1$, how to prove $a^x(1-ax)<b^x(1-bx)$? I've tried using $f(x)=x^t(1-xt)$ to do some manipulations (including derivations), but failed.
0
votes
2answers
40 views

need assistance identifying formula and help proving it

This is the identity: $$ \int_{x=0}^\frac{1}{\sqrt 2} \frac{x^{k-1}}{1-x^8}dx = \int_{x=0}^\frac{1}{\sqrt 2} { \sum_{i=0}^\infty x^{k-1+8i}}dx = \frac{1}{{\sqrt 2} ^ k} \sum_{i=0}^\infty ...
1
vote
1answer
26 views

Proving Monotonic Sequence Theorem

A sequence $b_n$ is decreasing and bounded. Prove it it convergent. Proof: Since $b_n$ is bounded, $b_n > L$ where L is the greatest lower bound as per the completeness Axiom. Consider some ...
0
votes
3answers
107 views

Find the problem with this proof.

The following attempts to prove that if $n^2$ is even, then $n$ is even. Suppose $n^2$ is even. Then $n^2$ = 2$k$ for some integer $k$. Let $n$ = 2$m$ for some integer $m$. This shows that $n$ is ...
1
vote
1answer
62 views

Show every dilation is a non-constant linear function.

A dilation of reals is a function $f:\Re \mapsto \Re$ such that for some constant $c\neq0$ one has $|f(x)-f(y)|=c\ast|x-y|$ for all $x,y\in\Re$. Show that every non-constant linear function is a ...
1
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0answers
50 views

Proof the Associative Property of H-group

I would like to prove the following proposition: Let $X$ and $Y$ be based topological spaces and let $[X,Y]$ be the set of homotopy classes of based maps $X\to Y$. If for every $X$, $[X,Y]$ is a ...
1
vote
1answer
60 views

Provide the Proof for $\forall x\,\bigl ( P(x) \land Q(x)\bigr ) \leftrightarrow \forall x \,P(x) \land \forall x\, Q(x)$

Provide the Proof for $$\forall x\,\bigl ( P(x) \land Q(x)\bigr ) \leftrightarrow \forall x \,P(x) \land \forall x\, Q(x)$$ This is all i got so far: Assume $\forall x \,\bigl( P(x) \land Q(x) ...
1
vote
1answer
62 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
46 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
1answer
48 views

When writing a proof, why do we want to assume a different but equivalent condition given in the proposition? [duplicate]

In the proof for the inductive step, we start by assuming $k \ge 10$. But along the way, the author mentions $k \ge 1$ and $k \ge 7$ to justify the inequality. Why do we bother to do this instead ...
0
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3answers
63 views

In proof by induction, what does it mean when condition for inductive step is lesser than the propsition itself?

My question is regarding the question posed at the end of the proof. My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false. ...
2
votes
2answers
100 views

Definition of a Limit

Prove that $ \ln\left(e +\frac{1}{n}\right) \to 1$ as $n$ approaches $\infty$. I know I must show $\exists$ $n > N$ such that $\left|\ln\left(e +\frac{1}{n}\right)-1\right|< \varepsilon $ But ...
0
votes
2answers
405 views

Proof that the subset relation is reflexive and transitive

I'm teaching myself set theory, and I'm not sure how detailed I should be when asked to prove things. Here is my proof that $A\subseteq A$ (the subset relation is reflexive): $A \subseteq B$ iff ...
0
votes
1answer
161 views

Base case in the Binet formula (Proof by strong induction)

I don't understand why the author says that the inductive formula does not apply until $u_3$. The formula does not depend on other terms in the Fibonacci sequence, so I thought I could just prove it ...
3
votes
0answers
58 views

An undirected graph $G$ can be decomposed into simple edge-disjoint cycles if and only if all of its vertices have even degree.

Research effort: $\rightarrow)$ I think this is relatively easy. $\leftarrow)$ Let $G = (V,E)$, let $w$ be any vertex of $G$, given that all the vertex have even degree, I'm assured that I can ...
1
vote
1answer
38 views

Prove that if sets A and B satisfy this relation, then they have a common element.

I have done the proof by drawing the picture and explaining it by using an example, but how could I start a more formal proof for this example without the use of a numeric example?
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2answers
32 views

Proving that mean KDR in a videogame is one

This is not related to schoolwork. A friend of mine challenged me to prove that the mean KDR (assuming players can only die at the hands of other players) must always be equal to one. I have gotten ...
1
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2answers
46 views

Is there a direct proof of the following?

I have been warned by my Lecture as well as several other sources that while proof by contradiction is useful and is certainly needed in some cases, it is often overused. In a effort to learn, I ...
3
votes
1answer
116 views

Euler proof of the formula involving factorial?

Let me be formal and write the formula Euler's Formula: Let $a$ and $n$ be nonnegative integers with $a\geq n.$ Then $$n!=a^n-\binom{n}{1}(a-1)^n+\binom{n}{2}(a-2)^n-\binom{n}{3}(a-3)^n+ > ...
1
vote
3answers
151 views

Proposed proof of set theoretic result

I am tasked with proving the following: $$ (A - B)\cap (B-A) = \varnothing $$ My Attempt: Suppose there exist a $x \in (A - B)\cap (B-A) $ then: \begin{align*} x \in (A - B)\cap (B-A) &\iff ...
3
votes
2answers
90 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 ...
3
votes
1answer
158 views

Prove that $2+2=4$.

Before you might chastise this quesion, I understand that we all know $2+2=4$. But a while ago I just stumbled across this paper which formally proves that $2+2=4$: ...
0
votes
2answers
75 views

Proof that doesn't exists a rational $s$ such that $s^2 = 6$

Well, I solved it, and I would like to know if there is anything that can be corrected or improved here. I think that the proof ended up too long, and with too many letters. Surely there is a better ...
4
votes
1answer
120 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 ...
-2
votes
2answers
85 views

Prove or disprove, Equivalence vs Implication? [closed]

Prove or disprove, for any universal set U and predicates P and Q [ ∃x∈U, P(x) ∧ Q(x) ] ⇒ [ ∃x∈U, P(x) ∧ (∃x∈U, Q(x)) ]
0
votes
2answers
75 views

Showing convergence rigourously

So I have $f_n(x) = x^{4n} + \frac1{n^2}$ which I know converges to $f(x)=0$ uniformly on interval $[0,1)$, but how can I show this with rigour? Is this acceptably rigourous? $\lim ...
1
vote
4answers
37 views

Lowest possible value of a function with derivative greater than 2

I have the following two problems, and want to attempt to solve them with Mathematical rigour(which I don't yet possess): Suppose that $f$ is differentiable on $[1,4]$ and is such that $f(1) = 10$ ...
0
votes
0answers
47 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
34 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
74 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
26 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: ...
2
votes
4answers
71 views

Prove that if product of matrices is singular, one of the matrices is singular.

I'm having trouble with this proof, it would be much easier to work out the other way it seems. Let $A$ and $B$ be square matrices of equal size. Prove that if $\det(AB) = 0 =C$ then either $A$ or ...
1
vote
1answer
33 views

Proof about isometries, symmetry and reversing orientation.

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an isometry of the reals. Prove that $f$ is a symmetry around a point if and only if $f$ reverses orientation of $\mathbb{R}$. The orientation of ...
2
votes
2answers
121 views

Proving the roots of a polynomial are irrational

This is a homework question so I'm just looking for some guidance. Basically we are asked to write a step by step proof in the form of assume/then statements for: $\forall x \in \mathbb{R}, ax^2 + ...
0
votes
2answers
43 views

Contrapositive proof using rule of divisibility

Suppose $x,y,z$ are integers and $x \neq 0 $ if $x$ does not divide $yz$ then $x$ does not divide $y$ and $x$ does not divide $z$. So far I have: Suppose it is false that $x$ does not divide $y$ and ...
0
votes
1answer
37 views

Proof about symmetry in isometries.

Suppose $f: \Bbb R \rightarrow \Bbb R$ is an isometry of the reals. Prove that $f$ is a symmetry about a point if and only if $f$ has a unique fixed point. Part 1: The assumption is $f$ is a ...
1
vote
3answers
36 views

Proof about isometries

Suppose $f\colon\mathbb R\to\mathbb R$ is an isometry of the reals. Prove $f$ is a non-trivial translation iff $f$ has no fixed points. Assumption: $f$ is a non-trivial translation (trivial ...
0
votes
0answers
50 views

finite subset topology

Let $X$ be a set and $\tau=\left\{X\right\}\,\cup\,\{u_i\subset X|u_i \text{ is finite}\}$. Is $\tau$ is a topology on $X$? My effort to show this is as follows: 1) $X\in\tau$ by definition and ...
2
votes
3answers
28 views

Help complete proof ('or' statement in conclusion): $a^2 \ge 7a \Rightarrow a\le0 \text { or } a\ge7$

Question Prove that if $a$ is a real number such that $a^2 \ge 7a$ then $a\le0 \text { or } a\ge7$ My Attempt We are given: $a \in \mathbb{R}$ $a^2 \ge 7a$ And need to prove: $a\le 0 \text{ ...