Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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$P (A_1)= P (A_1\cap A_2)+ P(A_1\cap A_2^c)$ (Proof verification)

Is my proof well writen and correctly stated? Let $(\Omega,\mathscr Z ,P)$ be a space of probability and $A_1,\dots,\in \mathscr Z$, then: $P (A_1)=P (A_1\cap A_2)+P (A_1\cap A_2^c)$ Dem: By ...
Roma_Rayado's user avatar
6 votes
5 answers
731 views

Is it possible to perform induction on the integers?

On a recent assignment, I had a question where I had to prove a certain statement to be true for all $n\in\mathbb{Z}$. The format of my proof looked like this: Statement is true when $n=0$ "...
June Richardson's user avatar
0 votes
3 answers
70 views

Proof that if A and B are connected sets, $A \cup B$ is connected iff $(\bar A \cap B) \cup (A \cap \bar B) \neq \emptyset$

The problem says: Let $(X,\tau)$ be a topological space so that $A,B \subset X$ are connected sets in said space. For $M\subset X, \bar M$ refers to the closure of $M$. Prove that $A \cup B$ is ...
Daniel Picazo Ezquerra's user avatar
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0 answers
50 views

A simple algorithm to find the inverse [duplicate]

Someone showed me such an algorithm. I'm going to explain his method with an example. Let's say we're looking for $$37^{-1}\in \mathbb Z/63\mathbb Z$$ First, we make the successive divisions of Euclid'...
Stéphane Jaouen's user avatar
0 votes
2 answers
62 views

Show that the sequence $(1+N^2)/[N(1+N)]-5/6$ is positive and increasing

I have the function \begin{equation*} f(N)= \frac{1+N^2}{N(1+N)}-\frac{5}{6}, \end{equation*} where $N\in\mathbb{Z}_{++}/\{1, 2\}$. I want to show that $f(N)>0$ and that $f(N+1)>f(N)$ for ...
Hal_Incandenza's user avatar
0 votes
1 answer
49 views

Struggling with proving an elementary atomic equation. Please critique my proof.

I am learning how to prove and disprove atomic equations. I have put together a proof where I know how I want to prove the theorem but can't quite get it to flow correctly. Please critique my syntax, ...
melon's user avatar
  • 19
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2 answers
112 views

Is there a visual proof for why this property of a matrix is true?

Let's say we have three equations written in their standard form: \begin{align} a_1x + b_1y + c_1 = 0 && (l_1) \\ a_2x + b_2y + c_2 = 0 && (l_2) \\ a_3x + b_3y + c_3 = 0 && (...
Fullk33's user avatar
  • 103
0 votes
0 answers
264 views

Proof verification: Every convex function on $(a,b)$ is continuous .

A real-valued function $f$ defined in $(a, b)$ is said to be convex if $f( tx+ (1 - t)y) \leq tf(x) + (1 - t)f(y)$ whenever $a < x < b, a < y < b, 0 < t < 1$ I want to Prove that ...
pie's user avatar
  • 3,299
2 votes
3 answers
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Proof that $\cos(x) = \cos\left(\frac{\pi}{2}\right)\implies x = \frac{\pi}{2} + k\pi$

I know that $$\cos(x) = \cos\left(\frac{\pi}{2}\right)$$ yields to $$(1)\qquad \qquad \qquad x = \frac{\pi}2 + 2k\pi$$ or $$(2)\qquad \qquad \qquad x = -\frac{\pi}2 + 2k\pi$$ which (as I found out in ...
ArziousYi's user avatar
0 votes
1 answer
30 views

Need hints (advice) to prove $(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$

I'm trying to prove this ( source : my uni's textbook says that it's trivial). $$(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$$ So far, I've managed to get ...
runtotherescue's user avatar
0 votes
0 answers
72 views

Difficulty with proving or refuting a claim related to a recursive sequence

I am trying to prove a certain claim about a recursive sequence. The sequence is defined as follows: $$ S_{n}^{k}(a) = \begin{cases} \sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a ...
Apollisto's user avatar
3 votes
2 answers
440 views

What is wrong with this proof by contradiction?

My professor tells me that a proof by contradiction structure below is poor: Proof. To prove: A implies B. So, assume A (in order to prove that B follows). Assume not B (in order to arrive at a ...
John Davies's user avatar
0 votes
0 answers
58 views

Prove that $(\forall a,b,c \in \mathbb{R}, b \neq 0 ) c\frac{a}{b}=\frac{a}{b}c=\frac{ca}{b}$ [closed]

Prove that $(\forall a,b,c \in \mathbb{R}, b \neq 0 ) c\frac{a}{b}=\frac{a}{b}c=\frac{ca}{b}$ The first equality can be solved using commutativity of multiplication, right? So: $c\frac{a}{b}=\frac{a}{...
runtotherescue's user avatar
0 votes
0 answers
41 views

When can a sequence be substituted for its limit

I'm new to proof-writing in Real Analysis, and came across the following problem: Prove $\lim(a_nb_n)=\lim(a)\lim(b)$. Let $\lim(a), \lim(b)$ exist and $\lim(a)=L_a, \lim(b)=L_b$ The first thing I was ...
Lambda's user avatar
  • 315
-1 votes
1 answer
83 views

In writing a proof how are we certain whether some part of a goal in the proof can be ignored or not? [duplicate]

I ask this question because I probably meet the same obstacle when writing other proofs. In writing a proof My example would be $(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$ The goal is $(0\le\sqrt{...
Stats Cruncher's user avatar
0 votes
1 answer
110 views

Are there attempts to use machine learning to settle disputes over long mathematical proofs?

As far as I remember, The proof of Perelman of the Geometerization Conjecture took a year to check by the experts. The claimed proof of the ABC conjecture Mochizuki took few years to check, and Peter ...
Amr's user avatar
  • 20k
0 votes
1 answer
115 views

Theorem 7, Section 4.5 of Hungerford’s Algebra

If $R$ is a ring with identity and $A_R$, ${}_{R}B$ are unitary $R$-modules, then there are $R$-module isomorphisms $$A\otimes_R R\cong R \text{ and } R\otimes_R B\cong B$$ Sketch of proof: Since $R$ ...
user264745's user avatar
  • 4,141
-1 votes
1 answer
99 views

Prove $(0\le a\lt b)\Rightarrow (0\le\sqrt{a}\lt\sqrt{b})$. Why is the proof of $0\le\sqrt{a}$ not needed?

$(0\le a\lt b)\implies(0\le\sqrt{a}\lt\sqrt{b})$ According to a proof by contradiction, Why is the proof of $0\le\sqrt{a}$ not needed? Proof: If $ (\sqrt{a}\ge\sqrt{b})$ then $a = (\sqrt{a})^2 \ge (...
Stats Cruncher's user avatar
0 votes
0 answers
43 views

Prove $AB=BA$ for all n x n matrices $B$ if and only if $A=\lambda I_n$ for some real number $\lambda$ [duplicate]

Trying to prove this but I'm stuck. I'm assuming A $\neq$ B. $$ (AB)_{i,j}=\sum^{n}_{k=1}a_{i,k}b_{k,j} $$ $$ (BA)_{i,j}=\sum^{n}_{k=1}a_{k,j}b_{i,k} $$ So we need $$ (AB)_{i,j} = (BA)_{i,j} $$ $$ \...
Beans's user avatar
  • 149
-4 votes
0 answers
51 views

Prove equality using only definitions for operations on sets. [closed]

Prove the following using only definitions for operations on sets: $A \cup B = A \oplus B \oplus (A \cap B)$ $A \times (B \setminus C) = (A \times B) \setminus (A \times C)$ I am clueless about how ...
Pipka Boykisser's user avatar
1 vote
0 answers
48 views

Proof Check: Let $(X,d)$ be a metric space and $\gamma:[0,1] \rightarrow X$ be a continuous function...

For the following question, is my proof correct? Can I write it any better? Let $(X,d)$ be a metric space and $\gamma:[0,1] \rightarrow X$ be a continuous function such that $\gamma(0) \ne \gamma(1)$....
Ethan's user avatar
  • 347
4 votes
1 answer
80 views

What are the conditions to compose limits to infinity?

I've recently become a little obsessed with all the ways limits can be composed - and the preconditions for this to take place. I took $A,B_{1},B_{2},C \subseteq \mathbb{R}$ and $f:A\to B_{1},\ g:B_{2}...
the thinker's user avatar
1 vote
0 answers
99 views

Which notation/abbreviation is appropriate to denote logical implication while writing proofs/solutions? [closed]

Find $x$ by solving the equation $$\frac{d (x^{2})}{dx} = 2$$ How should I write the remaining steps of the solution? Should it be $\implies 2x = 2$ $\implies x= 1$ OR i.e. $2x=2$ i.e. $x=1$ OR or $2x=...
Curiouser and curiouser's user avatar
0 votes
0 answers
43 views

Prove $F_{n} \leq \frac{7}{4}^{n-1}$ for all positive integers n.

Given the recursive definition of the Fibonacci numbers: $F_{n+1} = F_{n} + F_{n-1}$ $F_{1} = 1, F_{2} = 1, F_{3}= 2, ...$ I want to prove $F_{n} \leq (\frac{7}{4})^{n-1}$ for all $n \in \mathbb{N}$. ...
MattKuehr's user avatar
  • 121
5 votes
2 answers
115 views

Is this a valid proof that $A \cup (B \cap C) \not= (A \cup B) \cap C$?

I'm currently taking an introductory proof-writing course, and was wondering whether the following proof is incorrect: Prove or disprove the following statement: if $A$, $B$ and $C$ are sets then $A \...
Wabberjockey's user avatar
0 votes
0 answers
54 views

Show that $D(I_m + CD)^{−1} = (I_n + DC)^{−1}D$ [duplicate]

Knowing that $C$ is a member of $M_{m\times n}(F)$, $D$ is a member of $M_{n\times m}(F)$, and assuming that $I_m + CD$ is invertible, how would you show that $$D(I_m + CD)^{−1} = (I_n + DC)^{−1}D$$ ...
ms121399's user avatar
1 vote
1 answer
64 views

Show if $a$ has infinite order, then $a^i = a^j$ if and only if $i = j$

My abstract algebra textbook as the following part of a theorem that it almost treats as trivial, and I want to prove this myself: Theorem. Let $G$ be a group, $a \in G$ and $i, j \in \mathbb{Z}$. If $...
Mailbox's user avatar
  • 862
0 votes
1 answer
53 views

gamma distribution is to exponential distribution what ---- distribution is to Laplace distribution?

Context I'm working on a random walk problem with an two-sided exponential distribution (i.e., a Laplace distribution). From [1], I know that "The sum of $n$ independent $\operatorname{Exp}(\...
Michael Levy's user avatar
  • 1,050
0 votes
1 answer
44 views

Developing visual intuition for proofs involving cartesian product and sets

I am beginning to learn set theory proofs. It has been extremely useful to draw Venn diagrams for proofs just involving union, intersection, complement, e.t.c. However with cross product involved, how ...
tothemax's user avatar
2 votes
4 answers
674 views

Given a circle, construct three circles tangent to it and tangent to each other

Given a circle, construct three circles tangent to it and tangent to each other. Surprisingly, I can find no reference to this (although there are many similar sounding problems, and Apollonius' ...
SRobertJames's user avatar
  • 4,006
0 votes
2 answers
54 views

Peano axioms - do we need a specific property to show that the principle of mathematical induction implies the "correct" set of natural numbers?

From Terence Tao's Analysis I, Axiom 2.5 for the natural numbers reads My intuition behind this axiom is that every natural number is an element of a "chain" of natural numbers that goes ...
jvf's user avatar
  • 461
1 vote
1 answer
60 views

How do I approach this two column geometry proof?

I'm taking an online course and I'm a bit confused on what I'm supposed to do. I'm not looking for someone to do my school work for me, but if someone can explain to me how I should be approaching the ...
Ravenous's user avatar
  • 137
1 vote
3 answers
978 views

I think I just proved that Pi + e is irrational, but I know that is unsolved, so I'm wondering where the error in my proof is. [closed]

I had just finished some work on logic, and decided to attempt to prove some random propositions. I formed a proof by contradiction for e + π being irrational, the givens are: e is irrational, π is ...
IfFishThenSticker's user avatar
0 votes
0 answers
61 views

Prove that $\int^{x}_{0}{\frac{\sin(t)}{t+1}dt}>0$ if $x>0$ [duplicate]

Prove that $\int^{x}_{0}{\frac{\sin(t)}{t+1}dt}>0$ if $x>0$. This is problem 13.6 in Spivak's calculus textbook. Notably, I can't use FToC since that is in the next chapter. I have solved this ...
Nathan Rocha's user avatar
1 vote
2 answers
51 views

Algebraic Proof that the Amplitude does not affect Frequency

In Simple Harmonic Motion, the Amplitude is defined as the maximum displacement. Consider a pendulum (SHM) or a mass on a horizontal or vertical spiring. If the amplitude increases, which means ...
Kyotiq's user avatar
  • 39
0 votes
0 answers
50 views

Prove there are no natural solutions for an equation of two unknowns

I have the equation from a problem, for which I now need to show that the equation either has or doesn't have a pair of natural numbers, where $k$ is odd, as a solution: $$\frac{9}{40}x^2 + \frac{31}{...
user1192130's user avatar
7 votes
5 answers
677 views

Prove that if two polynomial functions have equal values over a closed interval, they are equal. [duplicate]

I want to prove that two polynomial functions that are equal over a specific interval, $(a, b) \in \mathbb R$ (closed interval with more than one point if that condition is necessary) are equal over $\...
Vector's user avatar
  • 343
1 vote
1 answer
46 views

To Prove that : P ∧ (Q ∨ R) is logically equivalent to (P ∧ Q) ∨ (P ∧ R). Is this a logical way of proving the statement?

I tried to prove the statement that P ∧ (Q ∨ R) is logically equivalent to (P ∧ Q) ∨ (P ∧ R). I have just started my first proof class so I apologize if the proof is bad/is repetitive. Please see my ...
Casemop's user avatar
  • 11
0 votes
1 answer
96 views

Prove that $\mathbb N \times \mathbb N$ is well ordered under $\le$

We define an ordering $\mathbb N \times \mathbb N$, $\le$ as follows: $(a, b) \le (c, d)$ iff $a \le c$ and $b \le d$. I tried to prove that $\mathbb N \times \mathbb N$ is well ordered under this ...
Vector's user avatar
  • 343
1 vote
0 answers
43 views

Thinking of a coordinate geometry proof on Pythagoras' Theorem with Euclid's idea but two squares only

Recently my colleague and I were thinking of a coordinate geometry proof on Pythagoras' Theorem with Euclid's idea but two squares only. But we are not sure whether it is an old idea or not and would ...
Psychomaths314's user avatar
0 votes
1 answer
77 views

Uncountable intersection of nondecreasing family of sets

Motivated by this question, consider a nondecreasing family of sets $(A_t)_{t\gt0}$, that is, assume that $A_t\subseteq A_s$ for every positive $t\leqslant s$. I want to show that for every strictly ...
S.H.W's user avatar
  • 4,379
3 votes
1 answer
93 views

Can we conclude that $f : (X,d) \longrightarrow (Y, \tau)$ is a homeomorphism?

Let $(X,d)$ be a metric space, $(Y,\tau)$ be a topological space and $f : X \longrightarrow Y$ be a bijective continuous map. Suppose that there exists $x_0 \in X$ such that for all $M \gt 0$ the ...
Anacardium's user avatar
  • 2,353
4 votes
1 answer
93 views

Proof of Positivity for Solutions in Ordinary Differential Equations (ODEs)

Let $x(t)$ be the solution of the initial value problem: $$ \dot{x}(t) = f(x(t)); \; \; x(0) = x_0 $$ I have made the following asumption during my work: If $x_0 \geq 0$, and $f(0) \geq 0$, then $x(t) ...
Olayo's user avatar
  • 67
1 vote
1 answer
100 views

Prove that if $x$, $y$, and $n$ are positive integers such that $x^{2024}$ + $y^{2024}$ = $2^n$ , then $x = y$.

Prove that if $x$, $y$, and $n$ are positive integers such that $x^{2024} + y^{2024} = 2^n$ , then $x = y$. My workings thus far: I have proved the case $n=1$, in which $x = y = 1$. For $n>1$, $2^...
Goat Man's user avatar
0 votes
1 answer
145 views

Is there way to prove that non-abelian group of order $p^3$ has non-trivial center using $G/Z(G) \cong Inn(G)$?

I know that $|Z(G)| \neq p^2$ or $p^3$ due to contradictions with $G$ being non-abelian. Therefore, the remaining possibilities are $1$ or $p$. I aim to prove $∣Z(G)∣=p$ without resorting to the class ...
Mahmoud albahar's user avatar
3 votes
1 answer
57 views

Greedy lemma Problem for matching spears to soldiers

The problem has $n$ spears and $n$ soldiers. Spears and soldiers have heights. We want to assign spears to soldiers such that the total height difference of spears and their assigned soldiers is ...
Yavuz Bozkurt's user avatar
4 votes
3 answers
149 views

Prove that every simple polygon has a ear without resorting to triangulation

We can establish the existence of a triangulation for each simple polygon relying on the fact that every simple polygon has at least one ear, utilizing induction. Conversely, we can establish that ...
log2's user avatar
  • 399
0 votes
2 answers
135 views

Can every statement that can be proved using the well-ordering principle be proved using weak mathematical induction?

The following is problem 30 of chapter 4.4 of Discrete Mathematics with Applications, 3rd ed. by Susanna Epp: Prove that if a statement can be proved by the well-ordering principle, then it can be ...
Cynicrom's user avatar
  • 327
0 votes
1 answer
77 views

half-closed nested interval property to prove $[0,1)$ uncountable

This rabbit hole led me to try proving $[0,1)$ a half closed-interval is uncountable via the Nested Interval property. I got distracted proving the bijection $(0,1)$ ~ $[0,1)$ and tried to prove both ...
Izak's user avatar
  • 167
-1 votes
0 answers
32 views

Proof of Backward induction

Can someone help me by checking/criticizing my proof for correctness and "style" of this Problem given in Taos Analysis 1? I would really appreciate some help as I have no one to check my ...
fwieland's user avatar

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