For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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1answer
10 views

proof detail concerning bijection between a set and its power set

Theorem: If $X$ is a set, then $X$ is not equivalent to its power set. Proof: suppose for a contradiction that $f:X\to P(X)$ is a bijection. Define $B:=\{x \in X, x\not\in f(x)\}$. Because $f$ is ...
0
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0answers
12 views

Riemann integrable function to the power of $p \in [1,\infty[$ is R-int. again.

It is stated that it is sufficient to prove Riemann-integrability of $|f|^p$ for $0 \leq f \leq 1$. $(f,\psi,\varphi:[a,b] \rightarrow \mathbb R)$. $\checkmark$ For any $\varepsilon >0$ there ...
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5answers
52 views

Demonstration of sum of powers of $2$ [duplicate]

Theorem : For every natural number $p$: $$\sum^p_{i=0} 2^i = 2^{p+1}-1$$ I trieed to demonstrate the theorem using induction Demonstration : $1)$ If we have $p=0$ then we get $2^0=2^{0+1}-1$ that is ...
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2answers
38 views

Prove/disprove $A\cap B=A\cap C $ for every $A$ $\iff B=C$

Let $A,B,C$ be sets, prove/disprove: $A\cap B=A\cap C $ for every $A$ $\iff B=C$ I think it's wrong, choose $A=\{1,2\}, B=\{2,3\}, C=\{2,4\}$ so $A\cap B=A\cap C$ but $B\neq C$ Although it's a ...
1
vote
2answers
26 views

Proving linear dependency for two vector groups

The question: Let V be a vector space over $\mathbb{R}$. Let $S = \{v,u,w\}$ be a group of 3 vectors in V. Let T be defined as $T = \{v, v + u, v + u + 2w \}$. Prove that if S is linearly dependent, ...
0
votes
1answer
26 views

Simple inequality proof in analysis

Just need verification on whether my proof is valid. I couldn't find a straightforward way to prove this inequality directly, so I tried a proof by contradiction instead. The question: Let $a, b \in ...
1
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0answers
20 views

Proving with a given definition that if $|A|=|B|$ then $A,B$ are equivalent (with induction but without using the induction hypothesis)

Let $A,B$ be finite sets, we'll say the sets are equivalent if $|A\setminus B|=|B\setminus A|$. Prove with the above definition that if $|A|=|B|$ then $A,B$ are equivalent. Suppose ...
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3answers
31 views

Proving if $p|ab$ then $p|a\vee p|b$, then $p$ is prime

Let $1\neq p\in \mathbb N$ such that $\forall a,b \in \mathbb N$ if $p|ab$ then $p|a\vee p|b$. Prove that $p$ is prime. My attempt, proof by contradiction: Suppose $p$ isn't prime, then ...
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2answers
33 views

Proving Pascal's identity

So I came across Pascal's identity: Prove that for any fixed $r\geq 1$, and all $n\geq r$, $$ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}. $$ I know you can use basic algebra or even an inductive ...
2
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3answers
53 views

Prove that $0 < 1$. Prove that $ab = 0 \implies a = 0$ or $b = 0$.

Proof: There exists $a = 0$ (For every $b$, an element of the set of positive numbers, such that: $b > a$) $$a + b > 0 \implies b > 0 \implies a < b.$$ Thus, we have shown that $0 < ...
1
vote
1answer
46 views

Is the solution to this elementary number theory problem correct?

Problem: A natural number $n$ is called nice if the following properties hold: • The expression is made ​​up of 4 decimal digits; • the first and third digits of $n$ are equal; • the second and ...
0
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3answers
35 views

Prove that if $l$ is a line in the classical euclidean plane, then there is a point $p$ that lies on $l$

Suppose that $\mathbb{P}$ is a Classical Euclidean Plane (satisfies all five of Euclid's postulates). Can you prove that if $L$ is a line in $\mathbb{P}$, then there is at least one point $p$ in ...
1
vote
1answer
20 views

Prove that for all $m$, there exist some $k$, such that $(m-n)^2 > m^2$ for all $n>k$

I have a problem where I need to prove: $\forall m \in \mathbb{N}:\exists m \in \mathbb{N} ∋(m−n)^2>m^2~∀n>k$ My thought was since it is only "there exists some k.." can I not say: if $k = ...
2
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0answers
28 views

Prove that $U_1\cup U_2$ is a subspace of $V$ $\iff$ $U_1\subseteq U_2$ or $U_2\subseteq U_1$ $\triangle$

Let $V$ be a vector space over some field. Let $U_1$ be a subspace of $V$. Let $U_2$ be a subspace of $V$. Prove that $U_1\cup U_2$ is a subspace of $V$ is equivalent to $U_1\subseteq U_2$ or ...
0
votes
1answer
27 views

Rectangles in one dimension

I have to prove the following proposition : Show that the intesection of two rectangles in $\mathbb{R}^{n}$ is either the vaccum or is another rectangle. My attempt: I one is embeded in the other ...
0
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0answers
29 views

Example: How to find inverse Laplace Transform by integral of the function (5.2-29)

This is just a demonstration on how to solve the following type of problem. Find $\mathcal{L}^{-1}\{\frac{54}{s^3(s-3)}\}$ by the given method: $$\mathcal{L}\{ ...
1
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2answers
44 views

Need help understanding Fibonacci Fast Doubling Proof

From this website, http://www.nayuki.io/page/fast-fibonacci-algorithms (fast doubling proof close to the bottom of the page). I have understood the proof for the most part but I am struggling to see ...
3
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4answers
53 views

Prove $\lim_{x\to2} (x^4 - 2x^3 + x + 3) = 5$ using Epsilon Delta

Prove $\lim_{x\to2} (x^4 - 2x^3 + x + 3) = 5$ using Epsilon Delta I am having difficulty finding the $\delta$ value. Here is what I have done so far: What I want to show: $$\forall \epsilon > ...
3
votes
1answer
39 views

Verifying a Vector Space Via Given Axioms

Let $X$ be the collection of all sequences $\{\alpha_n\}_{n=1}^{\infty}$ of scalars from $\mathbb{K}$ such that $\alpha_n=0$ for all but a finite number of values of $n$. Define addition and scalar ...
2
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1answer
38 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
1
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1answer
27 views

Proving that something diverges to infinity.

So I'm trying to prove that the sum of 1/(2k+1) diverges to infinity. I thought about doing a comparison test with the harmonic series 1/k and multiplying the harmonic series by (1/3) so it is (1/3k). ...
2
votes
0answers
32 views

Fiber product of non-abelian groups.

I am trying to understand whether surjectivity is needed for a fiber product of non-abelian groups to exist. I seem to have checked that the usual construction works for groups without any ...
1
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2answers
52 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 ...
1
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0answers
23 views

For what positive integer values $b,d$ does $(b^2-d)\mid(b^2-1)?$ hold?

I am curious about the answer to the following questions: And hope that you can help me For what positive integer values $b, d$ does $$(b^2-d)|(b^2-1)?$$ hold? Is it correct that the only ...
0
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1answer
22 views

Two example statements meant to demonstrate the importance of quantifier order don't appear to do so [duplicate]

In a book1 I have encountered the following: To check your understanding of [the importance of quantifier order], consider the following two statements. One is true, and the other is false. Which ...
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2answers
36 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: $\lim_{h \rightarrow 0} [f(x_{0} + h) - f(x_{0} - h)] = 0$ ...
2
votes
1answer
44 views

Proof Verification (Set Theory)

Let $S$ be a set with $N$ elements and let $A_1,\dots ,A_{101}$ be $101$ (possibly non disjoint) subsets of $S$ with the following properties: a) each element of $S$ belongs to at least one of these ...
1
vote
2answers
38 views

Induction proof concerning Pell numbers

Problem: The Pell numbers $p_n$ are defined by the recurrence relation \begin{align*} p_{n+1} = 2p_n + p_{n-1} \end{align*} for $n \geq 1$, together with $p_0 = 0$ and $p_1 = 1$. Prove with ...
0
votes
2answers
43 views

proving a limit by definition

given a sequence $a_n=a^{\frac{1}{n}}$ for $n\in\mathbb{N}^*$, $a\in\mathbb{R},a>1$ then proof that $\lim\limits_{n\to+\infty}a_n=1$ by definition. proof: given $a_n=a^{\frac{1}{n}}$ for ...
0
votes
1answer
23 views

AB = Identity matrix; matrices; determinants; proof

Let $M(n\times n, \mathbb Z)$ be the set of all $n\times n$- matrices with integer coefficients, and a matrix $A \in M$. Proof, that: There is exactly one matrix $B \in M(n\times n, \mathbb Z)$ with ...
0
votes
1answer
36 views

$\operatorname{rank}(A) = $max number of rows of submatrix $B$; Proof

I don't understand how to proof the following: The rank of a matrix $A \in M$ ($m \times n$, Field) equals the maximum number of rows of a square submatrix $B$ of $A$ with $\det (B) \neq 0$. The ...
0
votes
1answer
45 views

Elementary Operations on Sets

Let $X$ be a set with subsets $A$ and $B$. Prove: a). $X \setminus (X \setminus A) =A$. $X \setminus A$ is the set of all points of $X$ which do not belong to $A$. Given $p \in X$, we will show that ...
2
votes
1answer
40 views

$\sqrt{I}+\sqrt{J}=R$ implies $I+J=R$

Let $R$ be a commutative ring with unity and $I,J$ ideals of $R$. Suppose that $$ \sqrt{I}+\sqrt{J}=R $$ I want to show that this implies $I+J=R$. Take $r\in R$, then I can write $$ r=a+b, $$ for ...
0
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3answers
29 views

Proving if $a,b$ are even and $c$ is odd, then $ax+by=c$ doesn't have any solutions in $\mathbb N$

Let $a,b,c\in \mathbb Z$. Prove that if $a,b$ are even and $c$ is odd, then $ax+by=c$ doesn't have any solutions in $\mathbb N$. I get that sometimes this can acutally be false. Define ...
5
votes
3answers
61 views

Proving that if $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even.

Let $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$ then $abc$ is even. My attempt: If one or two numbers of $a,b,c$ are even then we're done, so we'll have to show that at least one of them is even. ...
2
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3answers
50 views

Is this a valid way to prove that $\frac{d}{dx}e^x=e^x$?

$$e^x= 1+x/1!+x^2/2!+x^3/3!+x^4/4!\cdots$$ $$\frac{d}{dx}e^x= \frac{d}{dx}1+\frac{d}{dx}x+\frac{d}{dx}x^2/2!+\frac{d}{dx}x^3/3!+\frac{d}{dx}x^4/4!+\cdots$$ ...
0
votes
3answers
46 views

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)

Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3) SO this is for abstract algebra and I am really struggling with this. Here are some of ...
0
votes
1answer
30 views

My proof that there are primitive roots modulo $p^2$

Let $p$ be a prime number. I'd like to prove that there are primitive roots modulo $p^2$. Could someone check this argument? Note that if $r\in\mathbb Z$ is a primitive root modulo $p^2$, it must ...
1
vote
1answer
13 views

Proof by induction that for a complete simple and undirected graph that $|E|=\frac {n(n-1)} 2$

Prove with induction that for a complete simple and undirected graph that $|E|=\frac {n(n-1)} 2$ Base case is trivial. Suppose that for a graph with $n-1$ vertices we have $|E|=\frac ...
3
votes
1answer
35 views

Axler LADR Exercise

The exercise is: Suppose $v_1, \ldots , v_m$ is linearly independent in $V$ and $w \in V$. Prove that if $v_1+w, \ldots, v_m+w$ is linearly dependent, then $w \in \operatorname{span}(v_1, \ldots, ...
2
votes
1answer
25 views

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational.

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
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votes
1answer
22 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?
1
vote
1answer
37 views

Showing that the class of all sets of a particular cardinality is not a set.

How to show that the class of all sets of a particular cardinality ,say $h$ is not a set. My argument: I assume that I've shown the following lemma. Lemma: If $X$ is an infinite set of cardinality ...
1
vote
1answer
37 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
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votes
0answers
37 views

Prove the inequality between the arithmetic and geometric mean

Assume that for $x_1,...,x_n\geq0$ we let $G=(x_1x_2\dots x_n)^{1/n}$ and $A=(x_1+x_2+...+x_n)/n$. I would like to know if the following procedure leads to a proof of $$G\leq A$$ The equality is ...
0
votes
1answer
48 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
1
vote
0answers
37 views

Inverse property for groups Proof

I was wondering if (1) this proof is correct, and (2) if other proofs exist for the following: Prove that $(a_1a_2...a_n)^{-1}=a_n^{-1}a_{n-1}^{-1}...a_1^{-1}$ where $a_i \in $ a Group $G$ Proof by ...
0
votes
2answers
24 views

Help with proof about functions and subsets

Problem: let $f: A \rightarrow B$. Prove that $f$ is injective if and only if for all $D \subset A$ we have that $f^{-1}(f(D)) = D$. Proof: => Suppose $f$ is injective. Let $x \in f^{-1}(f(D))$. ...
2
votes
0answers
55 views

Is my understanding of the argument correct?

I worked through a proof of: $$ f(z) = {1\over 2 \pi i}\int_{\partial D} {f(w) \over w -z} dw$$ where $D\subset \mathbb C$ is an open disk and $f$ is holomorphic on $D$ and continuous on ...
1
vote
1answer
28 views

Proof of the second principle of mathematical induction

This was an exercise in my lecture notes for which no answer was provided, so I seek verification on whether my proof is correct. Prove that if 1. $P(n_0)$ is true for some $n_0 \in \mathbb N$, and ...