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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2
votes
4answers
454 views

Proving that for any sets $A,B,C$, and $D$, if $(A\times B)\cap (C\times D)=\emptyset $, then $A \cap C = \emptyset $ or $B \cap D = \emptyset $

I'm trying to prove that for any sets $A$, $B$, $C$, and $D$, if the Cartesian product of $A$ and $B$ is disjoint with the Cartesian product of $C$ and $D$, then either $A$ and $C$ are disjoint or $B$ ...
2
votes
3answers
160 views

Prove that $(n, n + 1) = 1$ for all $n \gt 0$. [closed]

I was thinking of doing this by contradiction. So by supposing: $$(n, n + 1) \neq 1$$ Then trying to to show that $(n, n + 1) \gt 1$ or $(n, n + 1) \lt 1$. But I'm not sure how I can accomplish ...
4
votes
1answer
583 views

are two consecutive numbers relatively prime?

I have a question. I have been given this proof: "For any $n$ in the integers where $n>2$, show there are at least $2$ elements in $U(n)$ that satisfy $x^2=1$." I have gone through and actually ...
0
votes
1answer
89 views

The 'formal' way to prove equivalence?

In school we were always taught to prove equivalence by splitting an equation into $LHS$ and $RHS$ and working with each side individually until $LHS = RHS$. For example, prove: $$2^{k + 1} - 2 = ...
2
votes
0answers
120 views

Comparing Nash equilibrium and Pareto optimal actions

Suppose that $(x_{i}, x_{j})$ identify actions for two players $(i,j)$. If we define Pareto optimal actions by $$h(x_i) +h(x_j)- \eta[p(x_i)+p(x_j)]=2\gamma$$ and Nash equilibrium actions by ...
5
votes
2answers
125 views

Is this sloppy writing for limits?

Please note that I am not asking you to compute or show me how to do this limit. I am asking how to write out a clean and formal solution that is free of any error, ambiguity, or sloppiness. Given ...
0
votes
2answers
68 views

Help with math induction $S_{n+1}=1+aS_n$

Using mathematical induction I would like to prove the following. $S_0=1$ $S_1=1+a,...,S_n=\sum_{i=0}^n a^i$ Prove that $S_{n+1}=1+aS_n$ where $n \ge 0$.
3
votes
5answers
246 views

H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$

Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational? Where do I even begin?
0
votes
1answer
20 views

How does one select some $i$ to prove $\exists i$ s.t. $s_i \leq A$ where $A$ is the average of real numbers $(s_1 + s_2 + . . . + s_n) / n$

How does one select some $i$ to prove $\exists i$ s.t. $s_i \leq A$ where $A$ is the average of real numbers $(s_1 + s_2 + . . . + s_n) / n$ It seems like a trivial proof, but writing a proof for ...
5
votes
2answers
240 views

How do you get a paper to be peer reviewed

I have a proof that I want to undergo peer review. I unfortunately am not affiliated with any university. How should I go about getting it reviewed and either rejected or published? Thanks!
5
votes
4answers
5k views

Prove that $1^3 + 2^3 + … + n^3 = (1+ 2 + … + n)^2$

This is what I've been able to do: Base case: $n = 1$ $L.H.S: 1^3 = 1$ $R.H.S: (1)^2 = 1$ Therefore it's true for $n = 1$. I.H.: Assume that, for some $k \in \Bbb N$, $1^3 + 2^3 + ... + k^3 = (1 ...
1
vote
1answer
421 views

Show $\mathbb R^n$ is complete.

Show $\mathbb R^n$ is complete. At this point, I am trying to work through the problem in my textbook, there is one step that I do not understand and would like explained. Here's my proof so far: ...
1
vote
2answers
283 views

Is cancellation of Cartesian product possible? I think so, but am having trouble with the proof…

So, Is a cancellation possible for the Cartesian product? ex. if you have two Cartesian products that are equal to eachother, do the 2nd sets for each product equal eachother? Lets say you have ...
1
vote
1answer
525 views

Proving the sum of the first n natural numbers by induction

I have the Following Proof By Induction Question: $$ (1)(2) + (2)(3) + (3)(4) + \cdots+ (n) (n+1) = \frac{(n)(n+1)(n+2)}{3} $$ Can Anybody Tell Me What I'm Missing. This is where I've Gone So Far. ...
3
votes
3answers
99 views

it is surjective - $f: \mathbb{N} \rightarrow \mathbb{N}: x \rightarrow 2x$

i think, this function is surjective: $$f: \mathbb{N} \rightarrow \mathbb{N}: x \rightarrow 2x$$ but in my textbook it says, it is not surjective. but no proof there. i am really wondering if it is a ...
0
votes
2answers
137 views

math proof: $m\le n<m+1$, $n$ rational and $m$ positive integer. How do we prove $m$ is unique?

Suppose $$m\le n < m+1$$ $n$ rational and $m$ a positive integer. How do we prove $m$ is unique? I realize this is true if $m$ is an integer, so $m$ must be a unique number that is an integer ...
3
votes
4answers
1k views

Proving the sum of the first $n$ natural numbers by induction

I am currently studying proving by induction but I am faced with a problem. I need to solve by induction the following question. $$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$ for all $n > 1$. Any ...
1
vote
3answers
608 views

Show that there exists a positive real number $x$ such that $x^3 = 5$.

Here is what I've done so far: [First, want to show $b = 5$ is an upper-bound of $S$.] So, let: $$S = \{x \in \Bbb R : x \gt 0, x^3 \le 5\}, S \neq \emptyset$$ Assume that $b = 5$ is not an ...
2
votes
1answer
63 views

Show that $\exists \epsilon$ such that $f_{\epsilon}:=x+\epsilon g(x)$ is one-to-one where $g$ has bounded derivative

Consider a differentiable function $g : \mathbb{R} \to \mathbb{R}$ with bounded derivative $g'$, i.e. $\exists M>0$ such that $|g'(x)|\leq M$ for all $x\in \mathbb{R}$. Prove that for sufficiently ...
8
votes
1answer
333 views

Is “A and B imply C” equivalent to “For all A such that B, C”?

So I mostly study PDE, harmonic analysis, image processing, and so on, but for whatever reason I decided to be a TA for an undergraduate "introduction to proofs" course this semester. I suppose I ...
1
vote
2answers
122 views

Proof of the Sum of Square Roots

I have a question about a problem I encountered: $\exists$ a,b $\epsilon$ $\mathbb{R}$+ such that $\sqrt{a+b}=\sqrt{a}+\sqrt{b}$ Any tips for going about solving this? I tried: ...
5
votes
5answers
887 views

How does one begin to even write a proof?

I'm in my first proof based class and I'm just having a lot of trouble writing proofs. I mean I know it's not going to come natural and it will take time, but seroiusly, how does someone begin to ...
2
votes
2answers
98 views

$9^n \equiv 1 \mod 8$

I would like someone to check this inductive proof (sketch) The base case is clear. For the inductive step, it follows that $8 \mid 9^{n+1} - 9 = 9(9^n - 1)$ by the indutive hyp. So $9^{n+1} \equiv ...
0
votes
2answers
230 views

Vector span proof

Let $c_1,c_2,c_3,c_4,c_5$ be vectors in $R^4$ I'm trying to show that the set (call it set A) {$c_1,c_2,c_3,c_4,c_5$} spans $R^4$ if and only if the set (say, set B) of vectors ...
1
vote
1answer
119 views

Proving this language is not regular

Let $$L = \left\{b^ic^jd^k \mid i \ge 0, j\ge 0, k\ge 0,\text{ if }i=1\text{ then }j=k\right\}\;.$$ I have been trying to get a start on this proof for a long time now with no success. What would ...
0
votes
2answers
63 views

Proving a language is not regular

If two languages L1 and L2 are regular than so is their intersection. If follows that, for two languages L1 and L2, if L1 intersection L2 is not regular and L1 IS regular than L2 is not regular. I ...
1
vote
1answer
43 views

Counting Card hands with various restrictions

I would like to know if my solutions are correct for the following three combinatorial card questions. In each question, assume we have a standard deck of cards (13 ranks, and 4 suits). How many ...
0
votes
1answer
97 views

Simple proof involving inequality [duplicate]

Possible Duplicate: Proof with inequalities I've just started reading a book on real analysis and a lot of my proofs reduce to proving this fact over and over again: For all $\epsilon ...
0
votes
1answer
29 views

Help in showing probabilities are the same in Multinomial logit

I am using a multinomial logit to estimate $a_1, a_2, a_3, b_1, b_2$, and $b_3$ in a paper. My dependent variable takes 3 possible values, $y = \{1, 2, 3\}$; my independent variable is $z_i$; and the ...
0
votes
3answers
83 views

Finding K value to solve a problem using diophantine equation

I have to prove that any number that divided by 5 gives a remainder of 1 and divided by 7 a remainder of 2, also gives a remainder of 16 when divided by 35, using a diophantine equation. So first I ...
3
votes
4answers
142 views

Proof with inequalities

I have been assigned this problem for homework: Show that, if $a < b + \epsilon$ for every $\epsilon \gt 0$, then $a\le b$. I have tried to go about this using Induction, but I don't know ...
2
votes
5answers
985 views

Proof via Induction for A Summation

I'm starting to understand how induction works (with the whole $k \to k+1$ thing), but I'm not exactly sure how summations play a role. I'm a bit confused by this question specifically: $$ ...
1
vote
1answer
210 views

Nondeterministic finite automaton proof

I am having a really hard time working the problem below out. I am not sure I am even on the right direction with this logic . Swapping the accept and reject states alone is not sufficient to accept ...
1
vote
1answer
78 views

Special case of combinatorial onto functions

Let $[n]$ be the set of integers $\{1, 2, \ldots, n\}$. I want to find the number of onto functions from $[m]$ to $[3]$. The answer I found was $3!\cdot (3)^{m-3}$ My reasoning is: We have a ...
3
votes
1answer
134 views

Prove transitivity of $\forall X( X\subseteq A\setminus\{a, b\}\rightarrow(X\cup \{a\}\in\mathcal{F}\rightarrow X\cup\{b\}\in\mathcal{F}))$

This one is from Velleman's "How to Prove It, 2nd Ed.", exercise 4.3.23. Suppose $A$ is set, and $\mathcal{F}\subseteq\mathcal{P}(A)$. Let $$R=\{(a,b)\in A\times A : \text{for every } X\subseteq ...
1
vote
1answer
47 views

Combinatorial Correctness of one-to-one functions

Let $\lbrack k\rbrack$ be the set of integers $\{1, 2, \ldots, k\}$. What is the number of one-to-one functions from $m$ to $n$ if $m \leq n$? My answer is: $\dfrac{n!}{(n-m)!}$ My reasoning is the ...
4
votes
4answers
98 views

Am not following algebra in a proof - what am I missing here?

So I understand the majority of the proof, but am not fully following why consequently $n^2=9a^2$. Is this because we can take our value for $n$ (which is $n=3a$) and square it, which gives us $9a^2$? ...
1
vote
3answers
79 views

Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$

Have I justified that $\forall x \in \mathbb{R}$, $x > 1 \rightarrow x^2 > x$ Here is what I would do if this were asked on a test and I was told to "justify" the answer. Let $x \in ...
4
votes
2answers
158 views

Why does this step work in this proof?

I'm trying to learn discrete math and am brushing up on proofs by reading Richard Hammack's Book of Proof. I'm tripped up on this proof... I understand that it's contrapositive, and why contrapositive ...
5
votes
5answers
1k views

Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$

Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$ My first guess is to use an induction proof, so I started with n = k = 0: ...
3
votes
0answers
989 views

Proof of the sine rule

So I made my first attempt at a proof. I think it turned out well. Maybe not. But I was wondering if someone could take a look at it and tell me what they think. I'd be glad to hear some criticism on ...
2
votes
4answers
649 views

Proof: $x (y - x + 1) > y$ if $y > x$

I'm working on a computation which depends on the idea that given two natural numbers $x$ and $y$ where $y > x$, the product $x(y - x)$ will always be greater than $y$. Is there a proof of this ? ...
2
votes
0answers
565 views

Derivation of the density function of product of two random variables

I am looking for distribution of product of two random variables. Best I could found so far was this formula from the relevant Wikipedia page: $$ f_Z(z) = \int_{-\infty}^{+\infty} \frac{1}{|x|} ...
1
vote
1answer
27 views

How to get started on showing the conditions that $ax+by+cz=0$

I am looking at this question from Hardy's book, A Course of Pure Mathematics and have no idea where to begin. I was wondering, what is the first step to deriving the conditions? Question What are ...
1
vote
1answer
206 views

What is the difference between a binary relation and an equivalence class?

Is an equivalence class essentially a binary relation whose elements have an equivalence relation?
3
votes
3answers
192 views

Prove that the sets $(A\cap B)$ \ C and $(A\cap C)$ \ B are disjoint.

Please could someone validate this proof Prove that the sets $(A\cap B)$ \ C and $(A\cap C)$ \ B are disjoint. First we want to show that (1) $(A\cap B)$ \ C $\not \subseteq $ $(A\cap C)$ \ B ...
1
vote
1answer
99 views

Prove that $((p\lor q)\land(p\lor(\lnot q)))\rightarrow p$ is a tautology

Prove that $((p\lor q)\land(p\lor(\lnot q)))\rightarrow p$ Please could someone give me some feed back on this proof? Does it look correct? = $\lnot ((p\lor q)\land(p\lor(\lnot q)))\lor p$ = $ ...
2
votes
1answer
182 views

Growth of $ n^{\ln n}$ versus polynomial, exponential, and logarithmic forms

I'm attempting to clarify the proofs of these forms. Starting with $n^{ln\,n}$ I want to compare with polynomial, exponential, and logarithmic forms. I can understand just by looking at them which ...
1
vote
1answer
74 views

Eulers Approximation Inductive Proof

Alright so I am having some difficulty with an inductive proof. I am attempting to prove the following: Given that the Euler Method is described by the given recursive formula: $y_{n} = ...
5
votes
2answers
445 views

Can you prove why Popsicle Stick Multiplication works?

This is a unique way of multiplying numbers by using sticks. Let's call it "Popsicle Stick Multiplication". Or maybe "Linear Algebra" quite literally. Take a look at both images that I've drawn ...