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.

learn more… | top users | synonyms

0
votes
1answer
55 views

Proof using axioms of real numbers

Working on proof writing, and I need to prove $$(-x)y=-(xy)$$ using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am ...
0
votes
1answer
29 views

Proving a linear transform defined by an integral is injective

Let the fact that $I(p)(x)=\int_0^x p(s) ds$ is a linear transform from $P_4\rightarrow P_5$ be given. Prove that $I$ is injective. Would it be sufficient to just state that for any 2 ...
0
votes
2answers
31 views

Order of logical quantifiers within a statement

I understand that the order of the quantifiers of a statement determine the truth value of statement. For example, $$\forall x \in \mathbb{R}, \exists y \in \mathbb{R}\ \text{such that}\ ...
1
vote
1answer
61 views

Bounding the edges belonging to no perfect matching

We are told to let $G = (X \cup Y, E)$ be a bipartite graph with $|X|=|Y|=n$, and to suppose that $G$ has a perfect matching. I am trying to find a way to prove that $G$ has at most $n \choose 2$ ...
0
votes
1answer
34 views

Proving a linear transform is injective

Let $A:V \rightarrow W$ be a linear map. Prove that A is injective iff $\{v \in V :Av=0\}=\{0\}$ I read that a linear transform is injective iff the kernal of the function ...
0
votes
0answers
30 views

Proving greatest lower bound and least upper bound.

Could someone tell me how I can define glb and lub with logical/set symbols for the following question? I don't need complete proofs, I would like to know the givens and goals to work with here. ...
1
vote
3answers
47 views

Prove by induction $n^2 \leq n!$ for $n\geq 4$.

I managed to get $P(4):4^2 = 16 \geq 24 = 4!$ But then assuming $n^2 \geq n!, \forall n\geq4\in\mathbb{Z}$, I need to prove $(n+1)^2 \geq (n+1)!$ I tried $n^2+2n+1\geq n!\cdot (n+1)$, but I got ...
2
votes
1answer
36 views

Proof of linear independent eigenvectors

Hello I am looking for insight onto this theorem I will post. I will also post what I have for a potential proof, but I don't think it is very rigours. I am looking for one that maybe uses induction ...
1
vote
2answers
21 views

Prove that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses

Prove that if $f$ is a function such that $f'(x) > 0$ $\forall x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then $f$ ...
0
votes
2answers
46 views

How do I prove this? (Relations Proof)

So I can't seem to figure out how to prove this. Any help would be greatly appreciated. My professor said a contradiction would work but I don't see where I can make a contradiction. Show that {X ...
1
vote
3answers
31 views

Disproving a inequality implication by contradiction.

Let $x,y \in R$. If $0 \leq y < x$ for all $x > 0$, then $y=0$. Proof by contradiction: Assume the opposite that is; "If $0 \leq y < x$ for all $x > 0$, then $y\neq0$". ...
0
votes
1answer
11 views

Proving an existential quantified goal with a single universal quantified given

I'm trying to prove the following with no success: $\forall$A $\in$ F $\forall$B $\in$ G(A $\not\subseteq$ B) $\leftrightarrow$ $\cup$F $\not\subseteq$ $\cup$G In order to prove this statement, I ...
3
votes
4answers
65 views

Proving $ax+b$ is a linear function

$L\colon\mathbf{R}\to \mathbf{R}$ be given by $L(x)=ax+b$ over the scalar field $R$. I understand for that a function to be linear, it must adhere to the properties of additivity and scalar ...
1
vote
1answer
41 views

Bases and Galois Theory

I am working through the following notes: http://www.win.tue.nl/~sterk/algebra3/hoofd.pdf I have come across Proposition 2.4.7 on Page 21 which is given without proof. For completeness and clarity, ...
-1
votes
1answer
23 views

Showing that following is not a vector space?

I have the following 8 axioms for a Vector Space and the following question. I managed to prove that Axiom 3 doesn't work(and as a result Axiom 4 because 0 element doesn't exist) but the answer ...
0
votes
1answer
35 views

Proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous

Could you tell me some webpages or books where I can find the proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous. I know how it can proof if $P:\mathbb{R}\to\mathbb{R}$, but I don't know ...
1
vote
1answer
39 views

Use the principle of mathematical induction to prove that $n \lt (\frac{3}{2})^n$ for all integers $n \ge 1$.

Here's the problem: Use the principle of mathematical induction to prove that $n \lt (\frac{3}{2})^n$ for all integers $n \ge 1$. Here's what I've got: Base Case: $1 \lt (\frac{3}{2})^1$ is true. ...
3
votes
2answers
46 views

Continuous function with finitely many discontinuities is Riemann Integral

After a lecture today, I just wanted to confirm that I understand the proof of the following: If $f: [a,b] \to \mathbb{R}$ is bounded and continuous and has finitely many discontinuities, $f \in ...
0
votes
0answers
24 views

Prove that $D_{f_A,R}=\partial(A)\cup D_{f,A}$

Let $f:A\subset \mathbb R^n\to \mathbb R$ be a bounded function over bounded domain $A$ and let $f_A:R\subset \mathbb R^n\to \mathbb R$ $$f_A=\begin{cases} f(x), & \text{if $x\in A$}\\ 0, & ...
0
votes
2answers
25 views

Inverse Image Proof

Let $f:X\rightarrow Y$. Let $A$, $A_1$ and $A_2$ be subsets of $X$ and $B$, $B_1$, and $B_2$ be subsets of $Y$. Then, I need to prove that $f^{-1}(B_1\cup B_2)=f^{-1}(B_1)\cup f^{-1}(B_2)$. I know ...
0
votes
4answers
52 views

How do you algebraically derive “x <= 0” from “-x = | x |”

A = "-x = | x |" B = "x <= 0" If A, then B. By plugging in numbers or testing ranges less than zero, greater than zero, and equal to zero, I can verify that A ...
0
votes
0answers
50 views

Lebesgue theorem over jordan measurable sets

I want to prove that $f$ is integrable over a jordan measurable set $A$ iff the set of discontinuities of $f$ has measure zero, where $f:A\subset \mathbb R^n\to \mathbb R$ is a bounded function using ...
3
votes
1answer
72 views

Proof by induction: For all $n \geq 1$; $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots +(-1)^{n+1} \frac{1}{n} \leq 1$

Proof by induction: For all $n \geq 1$; $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots +(-1)^{n+1} \frac{1}{n} \leq 1$ This is what I have so far: Base case: for $n = 1$ ...
1
vote
1answer
33 views

Proof by induction: Show that for every real number $x\geqslant -1$ and every positive integer $n$, $ (1+x)^n \geqslant 1+nx$

Show that for every real number $x\geqslant -1$ and every positive integer $n$, $(1+x)^n \geqslant 1+nx$. This is what i have so far ...
0
votes
0answers
16 views

Proof Method and Absolute Value

I'm interested in the following method of proof for inequalities involving modulus : $( -x \geq C \wedge x \geq C ) \to ( |x| \geq C )$ $( -x > C \wedge x > C ) \to ( |x| ...
5
votes
3answers
46 views

Proving $\sum\limits_{k=1}^{2n} {(-1)^k \cdot k^2}=(2n+1)\cdot n$ for all $n\geq 1$ by induction

How prove the following equality: $a_n$:=$\sum\limits_{k=1}^{2n} {(-1)^k \cdot k^2}=(2n+1)\cdot n$ $1$.presumption: $(-1)^1 \cdot 1^2+(-1)^2+2^2=(2 \cdot 1+1) \cdot 1=3$ that seems legit ...
1
vote
0answers
24 views

Example of algebraic structure that is non distributive for BOTH distributive laws and how to do computation in them?

(Apologies if this one sounds like I have not done much research, or I did not aware already have an answer, but I have been searching everywhere and all of these structures presented here, even ...
1
vote
4answers
446 views

Suppose that x and y are irrational, but x + y is rational. Prove that x - y is irrational. [closed]

I can understand how it works in my head, I don't know how to prove it though.
0
votes
1answer
34 views

How do I prove that there exists a cyclic subgroup of order lcm of orders of cyclic subgrpups of an abelian group?

Before I start, please note that this post is not duplicate Let $G$ be an abelian group. Let $H,K$ be finite cyclic subgroups of $G$ such that $|H|=r,|K|=s$. Then, how do I prove that there exists ...
0
votes
0answers
10 views

Similarity of A and its transpose [duplicate]

Hello I am wondering if this theorem is true Theorem: Any matrix A is similar to its transpose, $A^T$ that is , there exists an invertible matrix P such that $A^T=P^{-1}AP$ I have a feeling that ...
6
votes
3answers
51 views

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$. I'm working towards figuring out an approach for finding that $|\sin{a} - \sin{b}| \leq ...
2
votes
2answers
22 views

Is this a valid step in a convergence proof?

I'm asked to say what the following limit is, and then prove it using the definition of convergence. $\lim_{n\rightarrow\infty}$$\dfrac{3n^2+1}{4n^2+n+2}$. Is it valid to say that the sequence ...
4
votes
1answer
89 views

Simple convergence proof

I'm asked to prove, using the definition of convergence, that limits approach a certain value. For example, $$\lim_{n\rightarrow\infty}\dfrac{n^2+4}{n^2}.$$ I can see that it converges to $1$, but ...
1
vote
2answers
22 views

Prove $d = \gcd(a,b) \iff 1= \gcd (k_1, k_2)$. [duplicate]

This is the assumption they give me: Let $a, b$ be integers and $d$ a positive integer. Let $d|a$ and $d|b$ so there there exists $a=dk_1$ and $b=dk_2$. I can go the backwards direction but I'm ...
9
votes
9answers
820 views

Advice on finding counterexamples

I am reaching out for specific advice on how one should go about finding counterexamples. It seems almost every time I've ever attempted a "find a counterexample" problem, I have to cheat by asking a ...
0
votes
1answer
16 views

Requirements to fully prove convergence using the integral test.

Based on the theorem below, when using the integral test to prove the convergence or divergence of a series, does one need to also prove the series itself is decreasing, continuous and positive? Would ...
11
votes
1answer
77 views

Valid proof of Young's Inequality?

Part of an exercise to prove Holder's inequality in Rudin involves proving Young's Inequality... That is, given $\frac{1}{p}+\frac{1}{q} = 1$, prove $$ab \leq \frac{a^p}{p} + \frac{b^q}{q}.$$ ...
0
votes
1answer
43 views

Is Proof By Induction Necessary? [duplicate]

Are there any theorems that can only be proved by induction? Induction seems to be proof by technicality.
0
votes
1answer
28 views

Proving that HK is a subgroup when K is normal

$HK = \{hk: h \in H\text{ and }k \in K\}$ I need to first prove first $e \in HK$. Since $e \in H$ and $e \in K$. Hence we have $e \cdot e = e \in HK$. Suppose $hk, h'k' \in HK$. $hk \cdot ...
0
votes
1answer
29 views

queuing problem, related to marriage algorithm

Say we have an nxn matrix and for every entry a_{ij}, it equals 1 if flight j starts after flight i ends. Otherwise it is 0. Suppose the largest matching contains M marriages (i.e. 1's in nxn matrix ...
0
votes
1answer
40 views

Proof by contradiction or contrapositive sets help

so I'm having difficulties proving the following Theorem, through either proof by contradiction or contrapositive. Can someone please help me? The problem is as follows: Prove that for any two sets, ...
0
votes
0answers
46 views

Prove that $Z(S_n)$ is trivial for $n \geq 3$ [duplicate]

Hi I am trying to solve this question I don't know where to begin but I have an idea so we have $$Z(G) = \{x \in G \space |\space xy = yx \space\forall y \in G\}$$ We must show that for all ...
0
votes
1answer
41 views

Iterating proof step

Many books proves theorems by performing one proof step and using this step as a scheme they say by repeating this step $l$ times we prove that... I wonder whether there is some formal meta-theorem ...
2
votes
1answer
47 views

Prove that $n(r) < 2\pi \sqrt[3]{r^{2}}$

Suppose that $n(r)$ denotes the numbers of points with integer coordinates on a circle of radius $r > 1$. Prove that $$ n(r) < 2\pi \sqrt[3]{r^{2}} $$ What process would you use to resolve ...
0
votes
0answers
23 views

Mixed strategies as LP problem

A row player is playing against a column player and his yield table is -, C1, C2, C3 R1, -3, 2, -1 R2, 0, -2, 1 R3, -1, 3, -5 Is it then correct to ...
0
votes
0answers
16 views

how to prove a non-negative integer n to be divisible by positive integer d is n mod d = 0

I'm not sure how to prove that, a necessary and sufficient condition for a non-negative integer n to be divisible by a positive integer d is that n mod d = 0. I get that I have to prove the cases of ...
1
vote
3answers
37 views

Mathematical Induction Proof Question dealing with integers

How would you use mathematical induction to prove that $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \dots + n \cdot (n + 1) \cdot (n + 2) = \frac{n(n + 1)(n + 2)(n + 3)}{4}$ I tried proving the base ...
2
votes
1answer
98 views

Proof that this specific function is measurable

Bounty Edit: Considering the nature of the problem at hand (i.e. proving that a specific function is measurable), I think this can be an easy but relevant problem. In particular, it is relevant to ...
0
votes
1answer
31 views

Proof approach: A 7x7 matrix with 15 ones can allow at least three marriages

This is quite difficult to prove imho with regards to Hall's Marriage Algorithm I can visualize a number of scenarios that work (i.e. put ones from the first entry to the fifteenth, or across ...
1
vote
1answer
51 views

Existence of functions $g$ such that 1. $f\circ g(1) =2$; 2. $g \circ f(1) = 2$, for all $f$ [closed]

Let $S = \{1,2,3,4\}$. Let $F$ be the sets of all functions from $S$ to $S$. a) Prove or disprove the statement: "For all $f \in F$, there exists $g \in F$ so that $(f \circ g)(1) = 2$" b) Prove or ...