For questions about the formulation of a proof, not about the mathematics behind it.

learn more… | top users | synonyms

0
votes
1answer
29 views

Probability Proof about A and B

I have to formally prove that: $$P(A) = P(A\wedge \neg B) + P(A\wedge B)$$ so I did like this: $$P(A\wedge \neg B) + P(A\wedge B)$$ $$=P(A\wedge \neg B) + P(A)\cdot P(B)$$ $$=P(A)\cdot P(\neg B) + ...
0
votes
0answers
17 views

Would this be considered a valid proof for $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$

I did a proof of the following $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$ using a proof by contra-positive, which was different from the direct proof that the solutions ...
2
votes
0answers
226 views

Theoretical proof of convergence of sequential weight update procedure (Neural Networks and Machine Learning)

My question is at the bottom. (Most of the descriptive words come from Chris. Bishop's Neural Networks for Pattern Recognition) Let $w$ be the weight vector of the neural network and $E$ the error ...
0
votes
0answers
24 views

Uniqueness of sum and multiplication of numbers

so I was writing a program that took two strings and said if they were anagrams or not, and I had this idea of making each character into a number, adding them all and checking if the result was the ...
0
votes
2answers
51 views

NEUTRAL GEOMETRY PROOF. prove that a figure can have at most one center of symmetry

A center of symmetry for a figure F is a point O such that every line through it cuts F in two points, P and P', such that O is the midpoint of PP'. Prove that a figure can have at most one center of ...
0
votes
1answer
491 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
-2
votes
0answers
21 views

householder transformation proof [on hold]

I'm not able to make a good proof which I understand of the following: Given $$\mathbf u\colon=\frac{\mathbf x+\rho\|x\|\mathbf e_1}{\|\mathbf x+\rho\|\mathbf x\|\mathbf e_1\|},$$ Prove that ...
1
vote
1answer
13 views

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$.

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$. My method: Let $y \in f(D) - f(E)$. Hence $y \in f(D)$ and $y \notin ...
2
votes
3answers
91 views

Prove that there is no largest irrational number

I have to prove that there is no largest irrational number from the result of the a previous proof: Prove that if $x$ is rational and $y$ is irrational then, $x+y$ is irrational. I was able to prove ...
0
votes
2answers
36 views

Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$

Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$. It was suggested to me that the easiest way to approach this was with a proof by ...
0
votes
1answer
71 views

What kind of proof is this?

Let's say that we want to prove that object A is blue. Is the following reasoning true? First assume that $A$ is indeed blue. Then, use other axioms to show that depending on a control parameter ...
23
votes
7answers
2k views

Can you use both sides of an equation to prove equality?

For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove ...
1
vote
4answers
35 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
1
vote
2answers
23 views

Let $S = { r \in \mathbb{Q} : r \lt 2}$. Prove that $S$ does not have a largest element.

Let $S$ = $[{ r \in \mathbb{Q} : r \lt 2}]$. Prove that $S$ does not have a largest element. My method: Assume to the contrary that $S$ does have a largest element, where $S$ = $[{r \in \mathbb{Q} ...
0
votes
2answers
11 views

Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies ...
0
votes
1answer
26 views

Proof for statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3

I am searching for a mathematical proof of this statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3. I have tried: To make a ...
3
votes
1answer
28 views

Prove $O(f(n)+g(n)) = O(f(n))$ when $g(n)=O(f(n))$

Given $g(n) = O(f (n))$, how can I prove that the following expression is true: $O(f (n) + g(n)) = O(f (n)) \tag1$ So I just write down what it says: $g(n) = O(f (n)) <=> f(n) \le c_1 ...
2
votes
1answer
36 views

Show that for propositional logic $\vdash_i \neg \varphi \Leftrightarrow \vdash_c \neg \varphi$.

As the title says, where $\vdash_i$ is derivations in Intuitionistic logic and $\vdash_c$ is derivations in Classical logic. I am allowed to use a corollary that states that $\vdash_i \varphi ...
0
votes
0answers
17 views

$\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ iif it exists $j\in\{1,…,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$

Show a vector $\vec c$ exists such that $\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ if and only if it exists $j\in\{1,...,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$ I'm only asking for a ...
1
vote
1answer
70 views

Correct process for proof in graph theory.

I'm working on what I'm sure is a fairly basic proof in graph theory. I must prove that $Every$ $graph$ $G$ $contains$ $a$ $path$ $with$ $\delta(G)$ $edges$. $\delta(G)$ is the minimum degree of the ...
2
votes
2answers
30 views

Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
1
vote
2answers
25 views

How to show that countable union of $F_\sigma$ is $F_\sigma$

On https://www.physicsforums.com/threads/countable-intersection-of-f-sigma-sets.666055/ Is it claimed that it is obvious that countable union of $F_\sigma$ is $F_\sigma$ Can someone elaborate why ...
0
votes
1answer
26 views

Proof formalization help: Given a vector $u$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at 2 points.

Proof formalization help: Given a vector $u$ of Euclidean length $1$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at at least 2 points. I've thought about the ...
1
vote
3answers
27 views

Proving a basis in linear algebra

So at the moment I'm trying to go through proofs and I came across this one: Suppose $ P_n $ is the vector space of all polynomials with degree less than or equal to n. Prove that $ \{1, x − 1, ...
0
votes
0answers
28 views

Proving a basis and dimension

Hi I'm currently looking at proofs in linear algebra and came across this one and I'm compketely baffled Suppose $ C_{ij} $ is the $2\times3$ matrix with $1$ in the $ i,j^{th} $ entry and zero ...
0
votes
1answer
26 views

How to use monotone convergence theorem to show that $\int \sum |f_n| = \sum \int |f_n|$

In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state $$\int \sum |f_n| = \sum \int |f_n|$$ without proof. Can ...
2
votes
1answer
24 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
0
votes
1answer
25 views

Clean proof for showing $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable

I wish to show that for $f:\mathbb{R} \to \mathbb{R}$, $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable Looks fairly easy if $f^{-1}([a,b))$ is one piece. Suppose ...
0
votes
2answers
43 views

How to prove that the square matrix $A_{n}$ matrix is nilpotent such that $A^{(n-1)}=0$

The matrix A looks like this: $$A=\begin{bmatrix} 0 & 1 & 0 & 0 & .&.&. &0\\ 0 & 0 & 2 & 0 & .&.&. &0\\ 0 & 0 & 0 & 3 ...
18
votes
1answer
779 views

Is there such a thing as a mathematical thesaurus?

I want this for two reasons: When writing proofs, I am constantly in need of synonyms of basic words like thus, there exists, for all, such as, contains, etc. A lot of mathematical concepts have ...
0
votes
1answer
28 views

Verifying multiplicative inverses of modulo n are the elements that are relatively prime to n

A proposition in my book states: $(\mathbb{Z}/n\mathbb{Z})^{\times} = \{a \in \mathbb{Z}/n\mathbb{Z}~|(a,n) = 1\}$ which I want to prove. I start by defining $a$ in terms of prime factors $$a = ...
0
votes
2answers
491 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
5
votes
2answers
72 views

Proof using induction: $n! > n^2$, for $n\geq4$

Proof using induction: $n! > n^2$, for $n\geq4$ Basis: For n = 4, we have: $4! > 4^2$ $24 > 16$ (TRUE) Inductive step: By the induction hypothesis: $k! > k^2$ $(k+1)k! > (k+1)k^2$ ...
2
votes
3answers
43 views

Proof help: Prove that $x^2+y^2+z^2 \geq xy+xz+yz$ [duplicate]

$x^2+y^2+z^2 \geq xy+xz+yz $ for all real numbers, x, y, and z. I'm not very good with working inequality proofs. Can someone help me prove this? The technique doesn't really matter.
0
votes
0answers
18 views

Existence of convergent sequence in a closed set?

Is it true that if $C$ is a closed set, and $x \in {C}$ then there exist a sequence $\{x_i\}$ which tends to $x$? I'm not sure about the correctness of this statement, and whether it is true for ...
1
vote
0answers
52 views

I did not understand one thing in the proof of substitution lemma?

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where x and y are ...
0
votes
3answers
44 views

Prove that graph with odd number of odd degree vertices does not exist

I need to prove that it is impossible to have a graph in which there are an odd number of odd degree vertices. What is the easiest way to formally prove this? I feel that I can prove it just by ...
-5
votes
3answers
96 views

Prove that $3^x$ divides $(3x)!\,\,\,\, \forall x \ge 1$ [closed]

Prove that $3^x$ divides $(3x)!\,\,\,\, \forall x \ge 1$ For example, $3$ divides $6 = 3!$
-5
votes
0answers
33 views

Proof by induction of for the cardinality of finite sets A and B [closed]

Can someone please help me with this proof? Proof by induction that for finite sets, A and B, an injection $f: A \rightarrow B$ exists if and only if A is finite and $|A| \le |B|$.
2
votes
2answers
25 views

Proof outer measure satisfies monotonicity: $A \subseteq B \implies m^*(A) \leq m^*(B)$

Theorem: $$A \subseteq B \implies m^*(A) \leq m^*(B)$$ Proof Attempt: By definition, $m^*(B) = \inf\{\sum\limits_{k=1}^\infty |J_k||\{J_k\} \text{ is a cover of B }\}$, $m^*(A) = ...
0
votes
2answers
30 views

Moments of a continuous random variable (Exercise 4.3.3 from Grimmett and Stirzaker)

Let $X$ be a non-negative random variable with density function $f$. Show that $$ E(X^r) = \int_0^\infty r x^{r-1} P(X > r)\,dx. $$ I tried using integration by parts to obtain \begin{align} ...
0
votes
0answers
17 views

Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k

Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k. I don't know how to start this one
5
votes
1answer
88 views

Is my proof correct? If $f$ has a finite number of discontinuities on $[a, b]$, then it is integrable on $[a, b]$

Question: Suppose a function $f(x)$ over the interval $[a, b]$ is bounded and has only a finite number of discontinuous points on $[a, b]$. I intend to prove that it must be integrable on $[a, b]$. ...
1
vote
1answer
26 views

Need help understanding algebra steps taken in proof of why an even minus an odd is odd

I don't understand the algebra used in the below example proof from my textbook. Where does the + 1 come from? Is it okay to just add 1 anywhere you want? Or is there some rule here or reason you ...
15
votes
3answers
612 views

Papers with unorthodox writing style

I'm not sure if this is the right forum for this question, in any case probably CW is appropriate? I've been looking around the mathblogosphere for the past few weeks and ran into mathgen. It's ...
0
votes
1answer
26 views

Proof of Interceting Lines

I have this practice problem from a final exam study guide. Let $f,g$ be continuous on $[a,b]$ and $f(a)>g(a)$ but $g(b)>f(b)$. Prove that $\exists c \in [a,b]$ such that $f(c)=g(c)$. My ...
2
votes
2answers
2k views

Is my proof correct? 'let a,b​​∈ Z. We write A | B if A divides B. Is the relation |, symmetric, transitive and/or reflexive?'

The relationship is not symmetrical. When a relationship is symmetrical: if xRy implies yRx for all x, y ∈ A (where A is a non-empty set, and R is a relation in A) If a, b ​​∈ Z, and as a | b means ...
1
vote
1answer
29 views

How to verify this relationship between area under the graph and the preimage?

Define $h : \mathbb{R} \to [0, \infty)$, Let $H = \{(x,y)| 0 \leq y \leq h(x)\}$ be the area under the graph (including the boundary) I wish to show the following is true: $$H = ...
2
votes
0answers
35 views

Simple Vacuous Proof, Correct Approach?

I am doing some practice exercises as I am starting out on proofs but I noticed that though I am getting the correct approach between vacuous and trivial proofs, I am not doing it in the same format ...
1
vote
1answer
16 views

How does one consider what a graph looks like in a mathematical proof

Mostly I am wondering for example what it would be like to prove that a linear graph (negative slope) shifted right would look the same as one shifted up. Can you consider how a graph looks when ...