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
76 views

Span and Smallest Submodule Proof

Let R be a ring, M a R-module, and $X \subseteq M$ Show that span$(X)$ is the smallest submodule of R containing X. My ideas: Every submodule is contained in its span so $X \subseteq$ span$(X)$ and ...
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0answers
55 views

2 column proof of The Tangent-Chord Angle Corollary

I need to prove 12.23 on this section (http://i.imgur.com/M5iev9K.png) I can use any of the theorems or corollaries before 12.23 but not the ones after it. This is a list ...
1
vote
1answer
471 views

Using Pigeonhole Principle for a graph proof

Using the Pigeonhole Principle, prove that in any graph with two or more vertices there must exist two vertices that have the same degree. (Note: the problem does not assume that the graph is ...
2
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2answers
208 views

Let a, b, c, d be rational numbers… [closed]

Let $a, b, c, d$ be rational numbers, where $\sqrt{b}$ and $\sqrt{d}$ exist and are irrational. If $a + \sqrt{b} = c + \sqrt{d}$, prove that $a=c$ and $b=d$.
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1answer
323 views

Proving the Division Algorithm using induction

Let $n \in \mathbb{N}$. For every $m \in \mathbb{Z}$, there exist unique $q, r \in \mathbb{Z}$ such that $ m = qn+r$ and $0 \le r \le n-1$. We call $q$ the quotient and $r$ the remainder when dividing ...
10
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0answers
244 views

My first proof that uses the well-ordering principle (very simple number theory). Please mark/grade.

What do you think about my first proof that uses the well-ordering principle? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof ...
2
votes
1answer
341 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
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vote
2answers
41 views

Ring with special rules for add and mult

$R$ is a ring of integers with special rules for multiplication and addition. Suppose that $f: \mathbb Z \to R$ is an isomorphism that is defined by $f(x) = x-2$. What integer in the ring $R$ is ...
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0answers
44 views

Show that the field Q[sqrt2] cannot be ordered using the defined relation

The complete questions states: On $\mathbb Q\:$[$\sqrt2 $] we define the relation: $\mathbb a+b\sqrt2 < a'+b'\sqrt2$ if $\mathbb a<a'$ and $\mathbb b<b'$ then show that the field $\mathbb ...
0
votes
1answer
52 views

GCD of polynomials

In $\mathbb{Z}/5\mathbb{Z}[x]$ use the Euclidean Algorithm to find the GCD of $x^4+x^2$ and $x^4+x^3+3x$. My thoughts: I am getting to the point where I need to use a fraction to get further in ...
0
votes
0answers
62 views

Let {$p_n$} be a sequence of points in the $\mathbb{R}^2$. Use the notion of convergence to solve the following

A) Define what it means for a point p $\in$ $\mathbb{R}^2$ to be a limit point of {$p_n$}. B) Prove that p is a limit point of {$p_n$} if and only if {$p_n$} has a subsequence which converges to p. ...
2
votes
1answer
109 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
1
vote
1answer
119 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
0
votes
4answers
57 views

direct proof of combination

Prove that $(^{n}_{2}) = 1+2+3+...+(n-1)=\sum^{n-1}_{k=1}k$ for $n \ge 2$ After some time flipping through notes I think I should use the sum of the 1st n natural is ...
1
vote
1answer
66 views

Building a partial injective relation

Question : A Partial Injective Relation from $A \rightarrow B$ is maximal if its graph of an injection function from $A$ to $B$ or the graph of an injection function from $B$ to $A$. Example: $A ...
1
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4answers
917 views

proof by induction: sum of binomial coefficients $\sum_{k=0}^n (^n_k) = 2^n$

Question: Prove that the sum of the binomial coefficients for the nth power of $(x + y)$ is $2^n$. i.e. the sum of the numbers in the $(n + 1)^{st}$ row of Pascal’s Triangle is $2^n$ i.e. prove ...
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2answers
29 views

Question with this proof

The integer $m$ is odd if and only if there exists q $\in \mathbb{Z}$ such that $m=2q+1$ I know that $m$ is even if 2|n, and $n$ is odd if $n$ is not even. I also know the division algorithm, which ...
3
votes
0answers
103 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
2
votes
2answers
73 views

Prove for a simple graph that $n-1 \leq m \leq \frac{n(n-1)}{2}$

For a given simple (that is neither loops nor multiple edges are allowed) undirected graph, where $m$ is the number of edges and $n$ is the number of vertices that the following inequality holds. ...
2
votes
0answers
64 views

My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade. [duplicate]

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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votes
3answers
835 views

My first induction proof (very simple number theory). Please mark/grade.

What do you think about my first induction proof? Please mark/grade. Theorem The sum of the cubes of three consecutive natural numbers is a multiple of 9. Proof First, introducing a predicate ...
0
votes
1answer
55 views

The limit of product of two functions, where one tends to infinity and the other to zero

If we know that $\lim_{x\to a}f(x)=\infty$, and $\lim_{x\to a}g(x)=0$, then what will be $\lim_{x\to a}(f(x)g(x))$? How do you prove the result? By using examples I can see that the limit can be ...
3
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0answers
134 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
3
votes
1answer
66 views

Functional form of a solution to a Differential Equation (Euler-Lagrange)

Let $f=f(q(t),\dot q(t),t)$, where $q(t)=\{q_1(t),...,q_N(t) \}=\{q_{a}\}_{a=1}^N$ and $\dot q:=\frac{dq}{dt}$. I want to show that if the following equations (Euler-Lagrange) are satisfied ...
0
votes
1answer
92 views

Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$.

Question: Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$. My attempt: By definition 6.23,6.3.1, and ...
2
votes
0answers
51 views

Help needed in understanding a proof

Claim: Let $M$ be a $R$-module ($R$ is an integral domain) and $p \in R$ be a prime. Suppose there exists non-empty finite subsets $B$ and $C$ of $M \backslash\{0\}$ such that $M= \bigoplus_{m \in ...
0
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1answer
44 views

If f(n)∈Ω(n) how do I prove or disprove f(n)∈O(n)

If f(n)∈Ω(n) how do I prove f(n)∈O(n) I feel it is true, but not sure how to show it the way I see it c1*n =< f(n) =< c2*n holds, but so confused on how to show it
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0answers
29 views

How to prove that $FC/FA + GC/GA= 0$ from this triangle problem?

In triangle $ABC$, a transversal line intersects $AB$, $BC$, $CA$ at $D,E,F$ respectively. $BS$ intersects $AC$ at $G$, where $S$ is the intersection of $AE$ and $CD$. How to prove that ...
1
vote
1answer
47 views

Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
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0answers
85 views

How to prove these equations base on this following interior and exterior angle bisectors problem?

In the triangle $\triangle ABC$, length of $BC$ is larger than length of $AC$. The interior angle bisector of $\angle C$ intersects $AB$ at $D$; and the exterior angle bisector of $\angle C$ ...
1
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1answer
122 views

Prove thoroughly: If the degree of all vertices is greater or equal to $\frac{|V| - 1}{2}$, then the simple graph is connected.

I am struggling to write a good, thorough proof. The proof is supposed to be logically rigorous, correct and complete (e.g. no hidden assumption). Moreover, style is important - the proof should be ...
0
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2answers
42 views

Should I create two distinct proofs? [*Soft question*]

This is a soft question, and if it is of poor quality, just let me know. As a method of improving my proofing abilities, should I make it habit to go about proving something twice. What I mean by ...
0
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1answer
53 views

Proof for a function $m:2^S\to R$

Let $S$ be a finite, non-empty set and $m:2^S\to R$ a function with the following properties $M1$: $\forall A\in2^S, m(A)\ge0$ $M2$: $\forall A, B\in 2^S, A\cap B=\varnothing\Longrightarrow m(A\cup ...
0
votes
1answer
25 views

Property of GCD in ring

Let D be a domain and $\emptyset \subset A \subseteq D^*$ Show that CD(A)={$d\in D$ | $(A)\subseteq (d)$} I know that I'll need to show both containments to show that the two statements are ...
0
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2answers
292 views

Combinatorics proof of “sum of (k choose m) with k from m up to n is equal to n+1 choose m+1”

I've already proved this statement algrebraically. I'm asked to prove it with combinatorics. So far I came up with, LHS= # ways to choose m apples from a total of m,m+1,...,n RHS= # ways to choose ...
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1answer
64 views

GCD Domain Proof

Let $D = \mathbb{R} + X \mathbb{C}[X]$ Show that $\gcd_D(X^2,iX^2)=\emptyset $ Here is my plan so far... (and my questions) Suppose $f \in \gcd_D(X^2,iX^2) $. How do I show that because X is ...
0
votes
1answer
32 views

Analytic, never zero on domain D graph and its local max. min

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum. Now, If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show ...
4
votes
7answers
979 views

How do I show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab?

I really need help with this question. I am required to show that the set of odd natural numbers is closed under the operation * defined by a*b=a+b+ab, and I'm not quite sure how. Any work/help is ...
2
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1answer
63 views

Beta function proof

Show that : $$\beta \left( x,n\right) =\dfrac {\left( n-1\right) !}{x\left( x+1\right) \left( x+2\right) ....\left( x+n-1\right) }$$ My attempt : $$\beta \left( x,n\right) =\dfrac {\Gamma \left( ...
0
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1answer
103 views

Proof: $ker(g) \subset ker(f)$ ..

Let $V = \mathbb{R}_{\le 3} [x]$ with basis $ B = (1, x, x^2, x^3)$. And $f: V \to \mathbb{R}, p \to \int_{-1}^1 p(x) dx$ and $g: V \to \mathbb{R}^3, p \to ^t( p(-1), p(0), p(1) )$. (1) I had to ...
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0answers
21 views

Show equivalence corresponding Nulls of function.

I'd like to show that the following two propositions are equivalent: (1) $f \in \mathbb{R}[x]$ has a multiple Null, so it's $\ge 2$. (2) $f$ and $f'$ have a common Null, whereas $f'$ describes the ...
0
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1answer
56 views

Proving two cosets are either equal or disjoint

I have a proof that I must produce, but I'm a little unsure of how to structure it, as I don't have a great deal of practice in forming rigorous proofs. We have, G the group of 2x2 invertible ...
3
votes
2answers
70 views

How to prove the formula of altitude from this following triangle?

Given: Right triangle $\triangle ABC$ with $A$ as right angle. If $t_A$ is altitude that drawn from point $A$ to $\overline{BC}$, called $\overline{AD}$. Prove that $t_A = ...
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0answers
36 views

Prove that $s \not= \emptyset $ by showing that at least one of $|a|$ or $|b|$ is an element of $S$.

. I'm trying to show an alternative proof to Bezout's Lemma (let $a, b \in \mathbb{Z}$, then there exists $x,y \in \mathbb{Z}$ such that $gcd(a, b) = ax + by$). Heres one of the steps in proving it: ...
1
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1answer
72 views

Cauchy's integral formula used on circle

If $\gamma$ is a piecewise, smooth, positively oriented simple closed curve in $D$, then Cauchy's formula states that $f(z)=1/2\pi i\int_\gamma {f(a)\over {a-z}}$. My textbook also stated that for ...
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2answers
112 views

Irreducible elements and Associates

Show that, in a domain, every associate of an atom is an atom. An atom is the same thing as an irreducible element. I think these two facts will be important to prove this statement: A nonunit ...
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1answer
60 views

How to prove that $DE=EF +DG$ from this following triangle problem?

Given a right triangle $ABC$, where $C$ is a right angle. We choose points $G$ at $AC$ and $F$ at $BC$, and $D$ and $E$ at $AB$. We draw right triangles $AGD$ and $EBF$, such that $\angle AGD= ...
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2answers
73 views

Having Trouble Forming Mathematical Proof

I'm having trouble forming a mathematical proof for a question. I can write down thousands of examples with various values of n that shows it's correct, but I'm not ...
0
votes
1answer
29 views

Possible metric space

$d_m$ is defined on $\Bbb R^2$ as such: $d_m(x,y) = max \lbrace|x_1 - y_1| , |x_2 -y_2| \rbrace $ where $x=(x_1,x_2) , y = (y_1 ,y_2)$ Which I have the task of proving whether or not the above s a ...
0
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2answers
49 views

Congruent iff Same Remainder (CISR) Confusion

I was reading through a proof of the following proposition: $ a \equiv b \mod{m} $ if and only if a and b have the same remainder when divided by m I came across a statement that I didn't quite ...