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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2answers
34 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
votes
1answer
49 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
16 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 ...
2
votes
0answers
28 views

Strict rules for determining whether defined objects are necessarily distinct - related to mathematical writing.

I am not a native speaker and I very often wonder whether there are rules (which are still unknown to me) for determining the exact meaning of mathematical statements/sentences. Currently, for many ...
0
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0answers
27 views

Need help on analytically proving the monotonicity of an inexplicit integral

I have the following function which I have numerically investigated to be monotonically increasing in $\nu$: $D(x;\mu,\nu) =\frac{1}{\sqrt{2\pi \cdot \nu}}\int_{-\infty}^{\infty}e^{-\theta \cdot x} ...
0
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0answers
32 views

Big-O and Big-Omega Proof

Trying to refresh my memory on proofs, I got a feeling this is true but can't remember how to prove it: If f(n)∈Ω(n) how do I prove f(n)∈O(n) Thanks for help (I do remember definitions of big omega, ...
0
votes
2answers
69 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 ...
0
votes
1answer
39 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
21 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
553 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 ...
0
votes
0answers
11 views

How to prove that angle EDL is same with angle ELD in the following triangle problem?

Given: triangle ABC with angle A equal with 60 degree. We choose points D and M at AC, points E and N at AB, such that DN perpendicular to AC and EM perpendicular to AB. If L is midpoint of MN, ...
2
votes
1answer
38 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
votes
1answer
48 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 ...
1
vote
0answers
20 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
votes
1answer
22 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
51 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 = ...
0
votes
0answers
28 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
vote
1answer
41 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 ...
0
votes
2answers
34 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 ...
0
votes
1answer
41 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= ...
1
vote
2answers
57 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
25 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
votes
2answers
20 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 ...
1
vote
1answer
36 views

How to show whether a statement is true or false(Example question inside)?

So I'm reading How to Read and Do Proofs by Solow and I'm on the exercises now. So far it has been good but I'm stuck on how to answer a question. There are no answers for even numbered questions in ...
2
votes
1answer
50 views

Metric Spaces: The dist function

Given that $A$ is defined as non-empty subset of $(X,d)$ The distance function is defined as such: $dist(x,A)=$ inf $_{y\in A} \lbrace d(x,y) \rbrace $ Given the above we are asked to prove the ...
1
vote
2answers
33 views

Show the equivalence: $ab|c \iff a|c$ and $b|c$

Let $a,b \in \mathbb{Z}$ \ {$0$} with $gcd(a, b) = 1$ and let $c \in \mathbb{Z}$. Show the equivalence: $ab|c \iff a|c$ and $b|c$ Also give an example of numbers $a,b \in \mathbb{Z}$ \ {$0$} and ...
0
votes
2answers
35 views

How would I prove for all a that a divides zero

I'm trying to prove for all a such that a divides zero. I can explain verbally why it works but I can't seem to be able to write it down in "proof" form. Could someone help me out?
3
votes
2answers
56 views

Proving the associativity of a monoid with $a \circ b = a+b-ab$

For university, it was my excercise to proof the associativity of the monoid $$H=(\mathbb{Q},\circ)\text{ with } a \circ b := a+b-ab\quad(a,b \in \mathbb{Q})$$ The excercise instructor gave us the ...
0
votes
1answer
24 views

Need help finding a formula for this sequence

A sequence $(x_j)^\infty_{j=0}$ satisfies $x_1=1$, and for all $m \ge n \ge 0 $ $x_{m+n}+x_{m-n} = \frac12 (x_{2m}+x_{2n})$. I have to find a formula for $x_j$ and then I can prove that later for ...
0
votes
3answers
63 views

Proving a function is a one to one correspondence

I understand that to show a function is a one to one correspondence, you have to show that the function is both one to one and onto. Proving a function is one to one seems simple enough. However for ...
0
votes
4answers
74 views

Show {$ ax + by | x, y \in \mathbb{Z}$} = {$n$ gcd$(a,b)|n\in \mathbb{Z}$}

I have the following problem: Let $a, b \in\mathbb{Z}$. Show that {$ ax + by | x, y \in \mathbb{Z}$} = {$n$ gcd$(a,b)|n\in \mathbb{Z}$} I understand that the Bezout's lemma says that $gcd(a,b) = ...
1
vote
1answer
38 views

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$.

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$. My Work: I considered the case where $n =-1$ , and the case $n \not= 1$. So when $n\not= -1$ we can let $n^2 - ...
1
vote
1answer
62 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
1
vote
1answer
34 views

does F(x) = -F(1/x) for all x in domain of F for this specific F(x)?

I apologize in advance for my ignorance on how to type mathematical symbols in this editor. Let F be the function defined for $x > 0$ by $F(x)= \int_1^x e^{((t^2)+1)/t}\frac{dt}{t}$ Show that ...
0
votes
3answers
25 views

Proving some number is a subsequential limit

Let $X_n$ be a sequence of real numbers. Suppose that for every $\epsilon>0$ and for every $m\in{N}$, there exists $n\geq m$ with $|x_n|<\epsilon$. Prove that 0 is a subsequential limit of the ...
0
votes
1answer
15 views

Question involving bounded sets and sequences

Let B be a bounded, nonempty subset of real numbers. Prove that there exists a sequence $X_n$ of real numbers such that for all $n\in{N},x_n\in{B}$ and $x_n\rightarrow\sup B$ My approach so far is ...
1
vote
1answer
53 views

Is there a simple way to prove the Four Colour Theorem?

The four colour theorem says that: Given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colours are required to colour the map so that no two ...
20
votes
5answers
3k views

Is a brute force method considered a proof?

Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$". If we let a computer check every single member of $X$ and conclude that the ...
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votes
0answers
20 views

Definition of Subrings

(5)The set $\{[0], [2], [4]\}$ is a subring of $\mathbb Z(6)$. I bieleve this is false. It is closed under add/mult but does it have a 0 or an x where a + x = 0 is satisfied in S? What does it mean ...
3
votes
1answer
68 views

Rings (integral domain and fields)

True or false: (1) Every integral domain is a field (2) every field is an integral domain (3) the ring $\mathbb Z$ is a field. (4) the ring $\mathbb Z/(17)$ is a field. (5)The set $\{[0], [2], ...
0
votes
0answers
41 views

Pascal's Triangle Proof problem

I'm having trouble with a Pascal's Triangle proof. I need to prove that if you start at any leftmost 1 (in a row i) in the triangle and take a diagonal of any length (i, 0), (i+1, 1), (i+j, j) you ...
0
votes
1answer
26 views

Validity of using “and so on” in a proof for finite number of iterations

I want to prove that for a metric space $(\frak{M},\rho)$ if $M\subset \frak M$ satisfies $$M\subset \mathop{\bigcup}_{k=1}^n B[x_k,r_k]\;\;(n\in\mathbb N),$$ then $M$ is a bounded set. Here's my ...
0
votes
2answers
19 views

Help with this proof (Index Sifting)

Let $(x_j)^\infty_{j=1}$ be a sequence in $\mathbb{Z}$ and let $a, b, r \in \mathbb{Z}$ such that $a\le b$. Then $\sum\limits_{j=a}^b x_j = \sum\limits_{j=a+r}^{b+r} x_{j-r}$ I assume that I would ...
0
votes
2answers
51 views

Struggling with a proof when $x > 0, x > a$, then $x > a$

This is coming from a question in spivak's calculus, solving $(x-1)(x-3) > 0$. There are two cases where this is true, when both brackets are positive, or when both are negative. But when I look ...
1
vote
0answers
57 views

Homework: prove there are two distinct integers $m$,$n$ such that $1/m+1/n$ is an integer.

I know this is Nth duplicate of this question, as I've seen it before, and I've been reading some of the answers as well. However, being really a total beginning with proofs, I'd appreciate if someone ...
1
vote
3answers
245 views

Prove that no number in this list is prime - Formatting a proof advice

Question: Let $n \in \mathbb{Z}$ where $n \geq 2$, prove no number in the list: $$n! + 2, n! + 3,...,n! + n$$ is prime. I have written my proof exactly as follows: Proof: $P(n) = n! + n = ...
1
vote
1answer
56 views

Proving the dual and self-dual of a poset.

The dual of a poset $(S \leq)$ is the poset $(S, \geq)$, where for all $x,y \in S$ $x \leq y$ if and only if $ y \geq x$. A poset is self-dual if it is isomorphic to its dual. Questions: Verify ...
0
votes
1answer
12 views

Diagonal Relation as a poset I - establishing the result by vacous truth

I have a problem with the logic behind the fact that, given a nonempty set X, the diagonal relation $D_X := \{ (x,x) : x \in X \}$ is a partial order on $X$. More specifically, my problem is with ...
1
vote
1answer
65 views

Proving very basic statements.

I'm just talking about (b), (c) and (d) in this question. The way I see it, (b) is asking to prove that: $$n \mod m = n \mod m$$which is like asking to prove that $1 = 1$. (c) is also asking to ...
1
vote
0answers
29 views

How can you prove the fundamental theorem of finitely generated abelian groups using the first isomorphism theorem?

I was able to prove the lemma that lets $G$ be a finitely generated abelian group, generated by $n$ elements $\{g_1,g_2,\dotsc,g_n\}$. Then the homomorphism $: \mathbb Z^n \to G$ defined by ...