For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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0
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4answers
50 views

GCD : Why does the GCD of two numbers divides their difference?

I was working my way through some number theoretic proofs and being a newbie am stuck on this proof : Why does the gcd of two numbers , say (a,b) - also divides their difference : a-b My ...
0
votes
1answer
41 views

GCD : (x,4) = 2 and (y,4) = 2 so is (x+y,4) = 4

I was working my way through some number theoretic proofs and being a newbie am stuck on this problem : If (x, 4) = 2 and (y, 4) =2, then (x + y, 4) = 4 where (a,b) denotes gcd of a & b ...
0
votes
3answers
34 views

Next step to take in this proof by contradiction?

This is a problem from Discrete Mathematics and its Applications Here is my work so far It's similar to this other question I had Next step to take to reach the contradiction?. I am assuming ...
1
vote
2answers
50 views

How to get to $5^3 \geq n^3$ in the proof by contradiction?

This is the same problem asked here. - Next step to take to reach the contradiction? Here is it again. I understand the solution - how you want to get to the fact 100 divides n^2 and then go ...
0
votes
1answer
41 views
+100

Conditional independence given elementary events implies conditional independence given $\sigma $-algebra

Proposition: Let $X$ be a continuous markov chain with discrete state space $S$. Let $Z$ be the corresponding jump chain and $\left\{ {{W_i},i \in \mathbb{N}} \right\}$ its holding times. Let ...
0
votes
0answers
22 views

How to prove that the integral of a positive, continuous function is positive?

Obviously intuitively the area under something that is above the x-axis is always positive, but how can I show this with a proof?
13
votes
3answers
763 views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
2
votes
2answers
45 views

Proving with induction $(1-x)^n<\frac 1 {1+nx}$

Prove using induction that $\forall n\in\mathbb N, \forall x\in \mathbb R: 0<x<1: (1-x)^n<\frac 1 {1+nx}$ My attempt: Base: for $n=1: 1-x<\frac 1 {1+x}\iff 1-x^2<1$, true since ...
2
votes
3answers
63 views

Can someone verify my direct proof that if $A$ is a subset of $B$, $A\cup B = B$?

This is a problem from Discrete Mathematics and its Applications I am trying to use a direct proof to do this problem. Here is my book's explanation/section on direct proof Here is my work so ...
0
votes
5answers
48 views

Number theory proof [on hold]

$(i)$ Prove that for every natural number $n \geq 2$, one has $(n + 1)|(n^3 + 1)$; $(ii)$ Suppose that $n$ is a natural number exceeding $1$. Prove that $(n^2-1)|(n^3+1)$ if and only if $n = 2$.
0
votes
1answer
30 views

prove that $p^2-1$ is divisible by $24$ if $p$ is a prime greater than $3$ [duplicate]

How to prove that $p^2-1$ is divisible by $24$ if $p$ is a prime number greater than $3$?
0
votes
1answer
29 views

Using set theory to prove a function problem

I begin with: $$A = \{a \le x < x_0 | f(x) = 0 \}$$ $$B = \{x_0 < x \le b | f(x) = 0 \}$$ Let $c = \sup A$ and let $d = \sup B$ First to prove $f(x) > 0$ for $x \in (c, d)$ I will ...
0
votes
1answer
26 views

Prove tautology without truth using a truth table. [duplicate]

I am struggling to prove, without using truth tables, that the statement is a tautology. [(p→q)∧(q→r)]→(p→r) My work so far... ¬[(¬p∨q)∧(¬q∨r)]∨(¬p∨r) ...
0
votes
2answers
72 views

Proof of a point's existence in an open interval

Well let us begin consider a set $A$ $$A = \{a \le x \le b \space | \space f(x) > 0 \}$$ Lets take $\alpha = \sup A$ and $\beta = \inf A$ What we must do is prove that $\alpha = d$ and ...
11
votes
8answers
247 views

What are the theorems in mathematics, proved using completely different ideas?

I know this question can have many answers. But I would like to know about theorems which can give completely different proofs. For example: I read from the book "Proof from the Book," there ...
2
votes
2answers
21 views

Homology groups of orientable surfaces.

I am trying to show that the second (simplicial) homology group or an orientable surface is ismormophic to $\mathbb Z$. I can show that this group is non-trivial by triangulating the surface, and ...
0
votes
2answers
32 views

Prove that $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$

Prove: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0})] = 0 \Rightarrow \lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ Proof: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ ...
1
vote
4answers
57 views

The divisibility of $a^p-1$ by $a-1$ and by $(a-1)^2$

I was working my way through some number theoretic proofs and being a newbie am stuck on this problem : Let a $\geq$ 2 and p be any positive integers , then prove that : $(a-1) \mid(a^p - ...
0
votes
2answers
49 views

What function to use to show one to one correspondence?

This problem is from Discrete Mathematics and its Applications Here's an example problem that the author gave I'am working on problem 2e. I first recognized the set as countably infinite. If you ...
2
votes
2answers
35 views

Show that the set of all cofinite subsets of S is enumerable.

I've been having some trouble with this question. In fact, I spend a long time on a solution which I came to realize the next day it was entirely wrong. I feel completely stumped, and I could really ...
1
vote
3answers
32 views

Proving by contradiction that if $a\in\mathbb Q,b\in \mathbb R\setminus \mathbb Q$ then $a+b\in \mathbb R \setminus \mathbb Q$

I'm trying to prove by contradiction that if $a\in\mathbb Q,b\in \mathbb R\setminus \mathbb Q$ then $a+b\in \mathbb R \setminus \mathbb Q$, I already proved it with contra position and a direct proof ...
7
votes
0answers
69 views

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
3
votes
1answer
22 views

How to proceed with the following integration?

If $n$ is a positive integer, show that $$ \int_{\sqrt{n\pi}}^{\sqrt{(n+1)\pi}} \sin(t^2) dt = \frac{(-1)^n}{c}$$ for some $c \in [\sqrt{n\pi}, \sqrt{(n+1)\pi}]$ I have an idea that i can use Mean ...
0
votes
1answer
26 views

Use the least integer principle to prove the following.

Least integer principle: Every non-empty set of positive integers has a least element. Using this fact, define $r$ to be the least integer for which $j - qk > 0$ where $j, k \in \Bbb{Z}$ ...
1
vote
1answer
29 views

How can I prove this about the tangent line formula??

The equation of a tangent line to $f(x)$ at $x = t$ is $y = f'(t)(x - t) + f(t)$. Recently, I heard that it is also determined by the remainder of polynomial division of $f(x)$ by $(x-t)^2$. For ...
1
vote
1answer
36 views

For how many integers is this a prime number?

For how many integers $n$ is: $9 - (n-2)^2$ a prime number? I want to try this using a rigorous definition of prime number/ actual problem rather than try-error? Please only give hints, so I can do ...
0
votes
1answer
38 views

Proving an “OR” statement

If one wants to proof $P\vee Q$, is it sufficient to proof $\lnot P \rightarrow Q$? Because it makes intuitively more sense to me that $P\vee Q$ would be logically equivalent with $(\lnot P ...
7
votes
6answers
3k views

Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$

I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. ...
0
votes
1answer
37 views

For what natural number $n$ is the following inequality true: $2^n \geq 2\cdot n^2$?

Can you solve this by using induction? The inequality is true for $n = 1$, but is false until $n = 7$. After the induction step I got $$2^n \geq n^2 + 2n + 1.$$ If you take the limit as $n$ ...
3
votes
1answer
97 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
0
votes
1answer
22 views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
-1
votes
1answer
19 views

question on proving inequalities [on hold]

If I need to prove $t(x) \ge0 $, for all $ x>0$ and I prove that $t(x) \gt 0 $, for all $ x>0$ does that make for a proof or is it wrong?
0
votes
1answer
61 views

What is the contraposive of this statement?

I have to prove the negation of this statement: $$\forall a,b,c\in\mathbb{Z}{\;if\;a\;|\;b\} $$ But the fact that there is a "and" is very disturbing. I think that I am missing something because my ...
0
votes
2answers
30 views

Olympiad minimum question, minimal value

If the numbers $A, B, C$ are such that the expression $\sqrt{A-B} + \sqrt{(B+3)^2} + C^2 - 4C + 4$ is as small as possible, then $A+B+C$ is? I thought start with, $A > B > C$ without loss of ...
5
votes
4answers
8k views

Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
3
votes
3answers
58 views

Rank in row echelon form

$$A= \begin{bmatrix} a & 1 & a & 0 & 0 & 0 \\ 0 & b & 1 & b & 0 & 0 \\ 0 & 0 & c & 1 & c & 0 \\ 0 & 0 & 0 & d & 1 ...
0
votes
2answers
43 views

Matrices that commute with all matrices [duplicate]

Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \ \lambda \in \mathbb R\}$$ ($I_n$ is the $n \times n$ identity ...
1
vote
1answer
45 views

Convert to Riemann Sum

The following limit has to be converted to Riemann Sum. $$\lim_{N→∞}\sum^{N}_{n=-N}\left(\frac{1}{(N+in)}+\frac{1}{(N-in)}\right)$$ My attempt: ...
1
vote
1answer
27 views

Linear Transformations: Proving 1 dimensional subspace goes to 1 dimensional

I am having trouble understanding this whole question, and how to prove it. Let $F:\mathbb{R}^n\to\mathbb{R}^m$ be a linear transformation. Prove that if $L$ is a $1$-dimensional subspace of ...
3
votes
1answer
56 views

Relationship between increasing integer sequences

Suppose that $\mathcal X\cap \mathcal Y=\emptyset$, that $\mathcal X\cup \mathcal Y=\Bbb N$ and that $X(n),\;Y(n)$ are increasing surjections $\Bbb N\to \mathcal X$ respectively $\Bbb N\to \mathcal ...
-1
votes
2answers
28 views

Prove that if $y>1$, then $\forall M\in\mathbb{R}$, there exists an $N$ in the natural numbers s.t. $n\geq N$ implies $y^n>M$. [on hold]

For $y\in\mathbb{R}$, prove that if $y>1$, then $\forall M\in\mathbb{R}$, $\exists N\in\mathbb{N}$ such that $$ n≥N \implies y^n>M. $$ I'm not used to proving these kinds of questions so any ...
0
votes
1answer
26 views

How to prove that the infinity norm of a matrix is the max of row sum?

I know how to prove that the 1-norm of a matrix is the max of the column sum, but not sure how to prove that the inf-norm is the max of the row sum. Any suggestion? Thanks
5
votes
2answers
371 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!
1
vote
2answers
100 views

Proving a function is onto?

Let $f: \mathbb{R}\setminus \{3\} \to \mathbb{R}\setminus \{1\}$ be defined by $f(x)=\dfrac{x+3}{x-3}$ Prove that $f$ is onto: Okay, here is the deal. I just started my first abstract algebra ...
0
votes
1answer
60 views

proof: $\sum\limits_{i=k}^n\binom{i}{k}=\binom{n+1}{k+1}$

Let $n ≥ 0$ and $k ≥ 0$ be integers. 1) How many bitstrings of length $n + 1$ have exactly $k + 1$ many $1$s? 2) Let $i$ be an integer with $k ≤ i ≤ n$. What is the number of bitstrings of length $n ...
-1
votes
1answer
16 views

Prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|))

For a,c, and x in the reals, prove that if a is not 0, then |x|>c>0 implies |(1/a)-(1/x)|<(|a-x|/(c|a|)). I'm trying to practice these kinds of questions, and any help or suggestions are greatly ...
3
votes
1answer
63 views

Could someone take a crack at this number theory problem?

The question is stated as follows: If $\mathrm{gcd}(a,m)=1$ and $X$ is a complete residue system $\bmod m$, then the set obtained by multiplying each member of $X$ by $a$ is also a complete residue ...
2
votes
3answers
114 views

Help with the algebra in for this number theory proof?

For all $n\geq 1$, prove with mathematical induction $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ So far.. I have substituted 1 and saw that the statement is ...
0
votes
1answer
27 views

Prove the associative law for the addition of real numbers

The problem asks us to prove the commutative and associative laws for the addition of real numbers. The commutative proof seems straightforward. I am wondering how to approach the proof of the ...