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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How to prove this statement and its negation?

Assuming that you're dealing with real numbers, d ^ 2 = e ^ 2, then d = e Why would it be true? << corrected, it is not true! thanks to posters What is the negation and is it true?
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72 views

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwsome number 1 to 17 that I forgotsolns.pdf There are exactly two nontrivial ...
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1answer
254 views

Characterize normal subgroups - Find all subgroups of $S_3$ conjugate to $\{id, (1,3) \}$ - Fraleigh p. 143 14.29

(27.) A subgroup H is conjugate to a subgroup K of a group G (viz. p. 141 $K \le G$ is a conjugate subgroup of $H$), if $i_g[H] = gHg^{-1} =K$ for some $g \in G$. Show that conjugacy is an ...
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1answer
287 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
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1answer
65 views

Show that $\frac{1}{n}X_n\to 0$ a.s.

Show that for any sequence $(X_n)_{n\in\mathbb{N}}\in (L_{\mathbb{P}}^2)^{\mathbb{N}}$ of identically distributed random variables it is $\frac{1}{n}X_n\to 0\text{ a.s.}$. The professor ...
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0answers
84 views

Please help me prove this thing. [too long to write all in the title]

"For two positive number a, b which satisfies the condition $\ln a \ln b <0$. equation $a^{b^{a^{b^x}}}=x$ has only one root if and only if ${\frac{d}{dx}a^{b^x}}_{x=t}\geq-1$, where $a^{b^t}=t$" ...
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2answers
149 views

Questions on proof. Every convergent sequence is bounded. (Abbott pp 45 t2.3.2)

(http://www.proofwiki.org/wiki/Convergent_Sequence_is_Bounded) Posit: $\{x_n\}$ is a sequence in $\mathbb{C}$. Posit: $x_n \to L$ as $n \to \infty$. Modus operandi. From the definition of ...
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4answers
736 views

Scratch work for delta-epsilon proof for $\lim_{x \to 13} \sqrt{x-4} = 3$

Prove $\lim_{x \to 13} \sqrt{x-4} = 3$. We need to show for all $E> 0$ there exists $D > 0$ such that if $0 < |x - 13| < D$, then $|\sqrt{x-4} - 3| < E$. Let me write D for ...
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1answer
33 views

Need help with this proof

The proof is as follows: Prove that $2 \neq 0 $, in the context of Natural numbers. Any help would be greatly appreciated. Thank you.
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179 views

If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers

Problem: If $\gcd(a, b) = 1$ and If $ab = x^2$ ,prove that $a$, $b$ must also be perfect squares; where $a$,$b$,$x$ are in the set of natural numbers I've come to the conclusion that $a \ne b$ and ...
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1answer
41 views

How to prove: there exists some $\lambda \in \mathbb{F}$ with $f=\lambda \text{Id}_{V}$

The problem says: Let $V$ be a finite-dimmensional vector spave over $\mathbb{F}$ and $f:V \rightarrow V$. Show that if with respect to all bases $f$ is represented by the same matrix $A$, then ...
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1answer
46 views

Centralizer Proof: $A \subseteq C(C(A))$

Show that $ A \subseteq C(C(A))$ Let $G$ be a group and $ A \subseteq G $. The centralizer of a subset of A is the set $C(A)=\{x\in G : ax=xa$ for all $a \in A\}$. *Isn't this obvious because $A=A$ ...
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2answers
130 views

Intuition — If $k \in \mathbb{Z}$ and $n \ge 2$, then the n$^{th}$ root of k is either an integer or irrational.

Origin — Elementary Number Theory — Jones — p25 — Exercise 2.4 (1) How do you prefigure the answer? Proofwiki start after prefiguring it. (2) What's the intuition? This answer ...
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2answers
131 views

Is the prove correct for: If both ab and a + b are even then both a and b are even

Show: If both $ab$ and $a + b$ are even, then both $a$ and $b$ are even Proof: Assume both $ab$ and $a + b$ are even but both $a$ and $b$ are not even Case1: one is odd $a=2m+1$, $b=2n$ Hence $a+b ...
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1answer
45 views

Prove this statement? [duplicate]

I am having trouble with the following proof: Prove that for every three positive real numbers a, b, and c that $(a+b+c)*(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \ge 9$. I have tried to directly ...
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4answers
459 views

Discrete Math Proofs Involving Real Numbers

I am stuck on these two problems. $1$. Prove that for every three positive real numbers a, b, and c that $(a+b+c)*(\frac{1}{a}+\frac{1}{b} + \frac{1}{c}) \ge 9$. $2$. Prove that for every three ...
2
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1answer
250 views

Combinatorial Proof -$\ n \choose r $ = $\frac nr$$\ n-1 \choose r-1$

I'm reading about combinatorics, specifically 'Cohen's Introduction to Combinatorial Theory', and am stuck on one of the problems. I'm looking for a combinatorial proof for the following : $\ n ...
2
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1answer
59 views

A question on rings

Let $R$ be an integral domain and $S$ be subring of $R$ with $1_R=1_S.$ Let $T=\{f(x) \in R[X]: f(0) \in S\}.$ Suppose $R[X]$ satisfies ascending chain condition for principal ideals, $ACCP.$ Could ...
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1answer
43 views

Proving that $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$

This assingment is preparation for exam. I need to prove with $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$ that $\sim$ is equivalence relatio. Can you tell me how to do this. ...
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1answer
163 views

Determinant divisibility by 9

I am having troubles with this homework problem, I tried expanding the determinant using co factor expansion but that didn't help. I know that for a number to be dividable by 9, the sum of its digits ...
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1answer
55 views

Prove that for all integers $n > 3$, $y_{n+1} = 2 x_n$

Let $x_n$ be the number of 0/1 strings of length $n$, not including the sequence 010. Let $y_n$ be the number of 0/1 strings of length $n$, not including 0011 or 1100. Prove that for all integers $n ...
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1answer
97 views

Prove $A $ \ $B $ = $A \cap B^c $

I see the use of $A $ \ $B $ = $A \cap B^c $ being used in bigger problems but how do you prove this? Is the proof as simple as: $A $ \ $B $ $\iff$ $ x \in (A \setminus B) \iff x\in A \cap ...
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1answer
48 views

Struggling with Proof Writing. Simple question for demostration.

I am practicing writing proofs over regular expressions. Here is the question: Show that $(r\cup \varepsilon)^*= r^*$, where $r$ is a string. Intuitively, the left hand side is the concatenation of ...
2
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0answers
118 views

Back-and-Forth Argument vs. “One-Way” Argument

The wikipedia article on the Back and Forth Argument claims at the end: If we iterated only step $(1)$, rather than going back and forth, then in some cases the resulting function from A to B ...
2
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1answer
46 views

If $\phi[H] \subseteq H'$, homomorphism from G to G' induces homomorphism from G/H to G'/H' - Fraleigh p. 143 14.39

Let $H \trianglelefteq \text{ group } G$ and let $H' \trianglelefteq \text{ group } G'$. Let $\phi$ be a homomorphism of G into G'. Show that if $\phi[H] \subseteq H'$, then $\phi$ induces a natural ...
2
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2answers
606 views

Proof for the formula of sum of arcsine functions $ \arcsin x + \arcsin y $

It is known that the following holds good: $$ \arcsin x + \arcsin y \\ \begin{align} &=\arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;\;;x^2+y^2 \le 1 \;\text{ or }\; x^2+y^2 > 1, xy< 0\\ ...
2
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1answer
490 views

Intersection of Normal Subgroups is Normal in Subgroup but Not Group - Fraleigh p. 143 14.35

Show that if H is a subgroup of a group G, and N is a normal subgroup in G, then $H \cap N$ is normal in H. Show by an example that $H \cap N$ need not be normal in G. I can condone the proof hence ...
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2answers
92 views

Symmetric Positive Definite and Gradient Proof

I have the function $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x - \mathbf b^T \mathbf x$ where $Q$ is symmetric. I'm trying to show that solving $\nabla f(\mathbf x) = 0$ is equivalent to solving $Q ...
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2answers
62 views

Struggling with a proof that $-m=(-1)m$

Prove that For all integers $m$, $-m=(-1)m$. Any help would be greatly appreciated.
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1answer
42 views

Proving Arithmetic

In mathematics if one is to prove a property of arithmetic, such as the associativity of addition, without going into greater detail about the numbers themselves, I feel like I'm missing something ...
2
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1answer
83 views

Showing $\sum_{n\in\mathbb{N}}\frac{\Lambda(n)}{n}=\log (x)+O(1)$ using a given result.

I'm stuck on the following problem. Use the fact that $$\sum_{p\le x}_{p\,\text{prime}}\frac{\log p}{p}=\log (x)+O(1)$$ to show that $$\sum_{n\le x}_{n\in\mathbb{N}}\frac{\Lambda(n)}{n}=\log ...
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1answer
87 views

Prove that every half-plane is a nonempty set.

So, I have no idea how to prove this. In my mind, I guess I'm thinking if you just had a line that made the half-planes and nothing else then why couldn't it be empty? If anyone can giveme an didea on ...
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1answer
113 views

Is proving both sides of iff necessary?

I have always been taught to prove both ways of an "if and only if" statement in a formal proof, but if the opposite way is very similar to the proof of the first way. Can you just leave a note and ...
0
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1answer
52 views

proof $B^{-1}MB$ is triangular

How to prove this? Theorem: Let $M$ be a matrix of complex numbers. There exists a non-singular matrix $B$ such that $B^{-1}MB$ is a triangular matrix. This is corollary from book Linear Algebra by ...
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3answers
173 views

Does $n!$ divide $ n^n$?

Today while I was reading on how to shuffle an array I came across a statement that claims we shall not swap an array entry with the whole array range when shuffling the array otherwise we end up with ...
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2answers
38 views

Formally proving that $\lim\left[ n^2/2^n \right] = 0$

Not sure how to formally prove this (specifically regarding the choice of $\epsilon$ in the formal limit definition)... Any suggestions?
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1answer
36 views

Greatest common divisor of multiples

What is the GCD of $3 \times 5^2 \times 7^2 \times 11^2$ and $3^2 \times 5^4 \times 11^3$? I can use the euclidean algorithm but is there an easier way to simplify this to make it more simple? If ...
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3answers
42 views

Proving primes divide each other

Suppose $a,b,p\in\mathbb Z$ with $p$ prime. Prove that if $p\mid a$ and $p \mid a^2 + b^2$, then $p \mid b$. I am starting with the fact that $a=p$t with $t\in\mathbb Z$ and $p= (a^2+b^2)\cdot x$ ...
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4answers
448 views

Fundamental theorem of arithmetic question

Let $b \in \mathbb{Z} $. Prove that if $p$ is a prime number such that $p | b^2$, then $p|b$. A certain theorem can be used to get this proof set up. I know the general rule that this scenario is ...
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1answer
86 views

Group Theory Exponent and Abelian Proof

Let G be a group such that $x, y \in G$ Show that, if $(xy)^2=x^2y^2$ or $(xy)^{-1}=x^{-1}y^{-1}$, then xy=yx. This can also be thought of as the exponent rule $(xy)^n$=$x^ny^n$ if xy=yx is true ...
2
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3answers
91 views

$n$-abelian Groups

Show that $(xy)^n=x^ny^n$ if $xy=yx$. I assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$. For the $n > 0$ case, can I use induction? For the base case I'll show that ...
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1answer
25 views

Proving by induction propositions of the type $P(n_1, n_2, …, n_k)$, where $n_1, n_2, …,$ and $n_k$ are natural numbers

For example: I've seen proofs of the multinomial theorem that use induction in the number of terms that are elevated at some power, but none that use induction in the exponent instead of using it in ...
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1answer
65 views

Proof of PDF Integrals

Hi guys my professor gave us some sample proofs to try at home and I was having trouble with 4 of them. I figured out how to do part (a) by using polar coordinates but cannot wrap my head around the ...
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1answer
88 views

Prove that if $V$ and $W$ are affine varieties, then $V \times W$ is an affine variety.

I am working on the problem from "Ideas, Varieties and Algorithms" by David Cox, John Little and Donal O'Shea. Here is the homework problem for my course. Let $V \subset k^n$ and $W \subset k^m$ ...
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1answer
51 views

Is This A Formal Proof $f:Z \rightarrow N ; f(z)=|z|$ is onto and not 1-1?

There will be a function $f(z)=|z|$ defined as follow $f:Z\rightarrow N$ Prove/Disprove the function is 1-1 or onto or both. Disproving 1-1 $f(-1)=f(1)=1 \rightarrow $-1 does not equal 1 therefore ...
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3answers
115 views

Question about the boudary of a set $A \subseteq \mathbb{R}^n $.

let $A \subseteq \mathbb{R}^n$. Let $X = \{ x \in \mathbb{R}^n : \forall \epsilon > 0, \; \; B(x, \epsilon) \cap A \neq \varnothing \; \; and \; \; B(x, \epsilon) \cap ( \mathbb{R}^n \setminus A ) ...
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1answer
31 views

About proof by induction

Proof by induction consists in following scheme: Proof by induction or intuitively, let be a predicate $ P(n) $ with $ n \in \Bbb{N} $: if $P(0) $ is true $P(k)\to P(k+1), \forall k \in \Bbb{N} $ ...
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4answers
73 views

$18 x \equiv 43 \pmod {23 }$.

Slightly similar to a question I submitted a few minutes ago, however this time my value for $x$ is $-602$, and I'm not sure how to write the general solution for $x$ .. help please? Help
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4answers
94 views

$74x \equiv 1 \bmod 53$

I've figured out that $x=-5$ for this to work but I don't know how to state the general solution? (all cases) Anyone know? D:
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2answers
69 views

Absorbing Element is a Unit

Show that an absorbing element of a monoid is a unit if and only if it is the only element. This is an if and only if proof so that means I have to prove it both ways: A implies B and B implies A. ...