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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Simple Proof question

Image : http://postimg.org/image/dkn0d5uen/ I'm studying Spivak's calculus and I have a really simple question : I'm only in the first chapter on "The basic properties of numbers" So far, we have ...
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1answer
84 views

Is this a correct proof by contradiction?

Prove that if a set $A$ of natural numbers contains $n_0$ and contains $k+1$ whenever it contains $k$, then $A$ contains all natural numbers $\geq n_0$. I have attempted a proof by contradiction as ...
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2answers
78 views

Proof that $\gcd$ divides $\operatorname{lcm}$

Show that the following conditions are equivalent: i) There exist positive integers $a, b$ such that $\gcd(a,b)=d$ and $\operatorname{lcm}(a,b)=m$. ii) $d\mid m$
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3answers
123 views

How do I derive $(\forall x)(\forall y)(\exists z)(x = y \circ z)$ from these three group axioms and some previously established theorems?

I am currently self-studying Patrick Suppes' Introduction to Logic and I am stuck on exercise 5.2.4. I've successfully worked out proofs for Theorems 1 to 7, but I am having trouble coming up with a ...
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4answers
253 views

Prove by induction that $10^n -1$ is divisible by 11 for every even natural number

Prove by induction that $10^n -1$ is divisible by 11 for every even natural number n. $0 \notin N$ Base Case: n = 2, since it is the first even natural number. $10^2 -1 = 99$ which is divisible by ...
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1answer
54 views

How can I formalise a proof of this?

A question from a textbook: Take the interval $(a,b)$, split into thirds likewise: $$(a,b)=\Big(a,\tfrac{1}{3} (2a+b)\Big)\bigcup \Big[\tfrac13(2a+b),\tfrac13 (2b+a)\Big]\bigcup ...
2
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1answer
205 views

Discrete Math Proof By Cases Confusion

I am currently finishing up my Discrete Math course, and I just wanted to clear something up that has confused me for the past few days. My teacher posts answer keys to assigned homework problems ...
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5answers
165 views

Prove that if $B-C$ $\subseteq$ $A^c$then $A \cap B \subseteq C$

Let A, B and C three sets. Prove that if $B-C$ $\subseteq$ $A^c$then $A \cap B \subseteq C$ Im trying to prove this with sheer logic and not making use of De Morgans laws etc. Let $y \in ...
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1answer
68 views

Let A, B, C and D be four sets:

Prove that if $A \cup B$ $\space\subseteq $ $\space C \cup D$, $A \cap B$=$\emptyset$ and $C\subseteq A$, then $B \subseteq D.$ I tried working around with this for a while and reached this ...
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6answers
301 views

Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers.

Let $m$ and $n$ be two integers. Prove that if $m^2 + n^2$ is divisible by $4$, then both $m$ and $n$ are even numbers. I think I have to use the contrapositive to solve this. So I assume $\neg ...
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1answer
76 views

Sum of $N$ natural numbers is less than $N+2$, then each number is less than $3$.

Prove by contradiction: If the sum of $N$ natural numbers is less than $N + 2$ then each of these numbers is less than $3$. Attempt: I have to assume that the sum of $N$ natural numbers is greater ...
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1answer
203 views

Why are proofs written in first person plural? Were they ever written differently?

It's probably a silly question but it interests me when was the convention of writing proofs in first person plural introduced? Is there any historical examples of a different POV for proof writing?
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1answer
62 views

Is there a rule to justify the following logical statement?

I have to derive the following expression and reach the second one: $$\begin{gather} ( ( \forall x , Q \Rightarrow \neg P (x) ) \wedge ( \forall x, \neg Q \Rightarrow \neg P(x) ) ) \\ \iff \\ ( ...
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4answers
58 views

Prove that $A=B$

Let A and B be two subsets of some universal set. Prove that if $(A ∪ B)^c$ = $A^c ∪ B^c$, then $A=B$ ATTEMPT: Let y $\in (A ∪ B)^c$ this means that $y\notin A$ or $y\notin B$ which is equal to $A^c ...
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1answer
72 views

Does proving that a function is not in big O mean that the function is in big Omega?

If I determine that a function is not in Big O of another function, can you assume that the function is in big Omega of the same function?
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1answer
75 views

Gauss-Bonnet theorem question

I was wondering if someone here can give me a hand with the proof in the image below. This is not HW, just a brain-teaser I am working on. Prove. $2\pi \chi(M)=\sum\limits_{v_i}k(v_i)$
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1answer
105 views

Formal Definition of Limit and Proofs

I'm having trouble understanding the formal definition of a limit... Let $f(x)$ be defined on an open interval about $x_0$, except possibly at $x_0$ itself. We say that the limit of $f(x)$ as ...
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1answer
73 views

Models and their meaning in a proof of any formula

Behind the scenes all formula $\phi$, we must define a model, M = (F, P) over a Universe, where F = set of Functions and P = set of Predicates, on a table of free variables in $\phi$ ? Ie any $\phi$ ...
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2answers
68 views

Cantor-Schroder-Bernstein Contradiction

I need help figuring out where to start a proof that says I should use a proof by contradiction. $f\colon A\to B$ and $g\colon B\to A$ be functions and each is 1-1. Let $D$ be the range of $f$ (i.e., ...
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1answer
52 views

Every two consecutive integers are coprime.

I start with knowing that two numbers are coprime if: $n*k + m*j = 1$ So, setting $k = a$ and j = $a+1$ I can solve as follows: $n*a + m*a + a$ Then, $a(n+m) + m = 1$ Where can I go from here?
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3answers
133 views

What to look for in a proof?

I am a physics undergrad, wishing to pursue a PhD in Math. I am mostly self taught in the typical math undergrad curriculum. I am looking for more input, in ways I can improve my mathematical ...
0
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1answer
82 views

Proof polynomial is always divisible by number

Given $f(x) \in \mathbb{Z} [x] $ a polinomyal, that evaluated in any $a \in \mathbb{N} $, results allways in a multiple of 101 or a multiple of 107 (both prime numbers). Prove then, that $f(x)$ it's ...
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1answer
74 views

Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that… [duplicate]

Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that $$\lim_{x\to 0}\bigg(\dfrac{a_1^x+a_2^x+\cdots+a_n^x}{n}\bigg)^{\frac{1}{x}}=\sqrt[n]{a_1a_2\cdots a_n}$$ Got no clue where to begin from ...
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1answer
28 views

Prove Bijection in roots of unity function

Given $k \in \mathbb{N}, G_k = \{z \in \mathbb{C} |z^k =1 \} $. Probe that if $n$ and $m$ are coprime, the function $f: G_n \times G_m \rightarrow G_{mn}, f(\alpha, \beta) =\alpha\beta$ is bijective. ...
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1answer
60 views

Let $x$ be a real number. To prove…

Let $x$ be a real number. Define the sequence $(x_n)_{n\ge1}$ recursively by $x_1=1$ and $x_{n+1}=x^n+nx_n$ for $n\ge1$. Prove that, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
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1answer
55 views

Question about convergence of a power series and when the series is not zero

Following is a past exam question I am trying to solve as a preparation for my own exam. Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers with $a_n \leq M$ for some $M\in\mathbb{R}^{+}$ and ...
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6answers
312 views

How to explain that proof is important

I don't know if this is the right place to post this or not, but I will go ahead anyway (sorry if it ain't the right place) Yesterday I was discussing a particular theorem of geometry with my brother ...
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3answers
67 views

proof by contradiction $A ∩ (B - A)= \varnothing.$

Use the method of proof by contradiction to prove that if A and B are sets, then $$A ∩ (B - A)= \varnothing.$$ It says I have to use contradiction, but contradiction is the one I have a problem with! ...
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1answer
67 views

Linear algebra hw! Linear transformation

Let $T : V -> V$ be a linear transformation where V is a nite dimensional vector space. If rank(T) = rank$(T^2)$, prove that image(T)$\cap$Ker(T) = {0}. I have to give this hw to my prof this ...
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1answer
103 views

Prove or disprove: If $A$ and $B$ are denumerable, then $A - B$ is denumerable

Prove or disprove: If $A$ and $B$ are denumerable, then $A - B$ is denumerable Can someone give me a hint as to how to prove/disprove this? My instinct tells me that the claim is true. But I'm ...
3
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3answers
123 views

Proof for $\sinh(x-y)$

Basically I need to prove that $\sinh(x-y)=(\sinh x)(\cosh y)-(\cosh x)(\sinh y)$ I could use the fact that $\cosh$ is an even function and $\sinh$ is an odd. I can prove that: $$\sinh(x+y) = \sinh ...
4
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1answer
93 views

Combinatorial Identity Proof

What is a combinatorial proof for this identity: $1 \times 1! + 2 \times 2! + ... + n \times n! = (n + 1)! - 1$ I am trying to figure out what are both sides trying to count.
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4answers
65 views

Prove that $S = \left \{(x, y) \in \mathbb{N} \times \mathbb{R}: xy = 1 \right\}$ is denumerable

In the solutions, the proof begins by defining the function $f : S \rightarrow \mathbb{N}$ by $f(x,y)=x$ and goes on to show that $f$ is a one to one correspondence from $\mathbb{N}$ to $S$. However, ...
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3answers
74 views

Prove that if $A \subseteq B$, then $A \cup C \subseteq B \cup C$

Assume that $A, B, C$ are arbitrary subsets of $\mathbb{N}$. Prove that if $A \subseteq B$, then $A \cup C \subseteq B \cup C$ Proof: Asume that $A \subseteq B$, then there exists an $x \in ...
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2answers
46 views

Math Proof Help

So I am supposed to add some condition to the original proposition to make it true but I do not know what condition I need to add. Original Proposition: If $x$ and $y$ are real numbers and $xy>0$, ...
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1answer
126 views

Logic Proof: Help Starting out

I am having some trouble starting off this proof. I am not sure if I need to prove by the contrapositive or if it is a direct proof. Prove: If $x, y,$ and $z$ are natural numbers such that ...
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2answers
118 views

learning how to write proofs properly

I am learning how to structure my proofs in such a way that others can read them with ease. It was pointed out to me several times on this site that my proofs are not very clear. Anyway, here goes: ...
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1answer
44 views

prove two different forms of the same uniqueness theorem are logically equivalent

One may take either of the statements below as a definition of $(\exists!x)(P_x)$, where $P_x$ is a predicate concerning the set $x$. Prove that they are logically equivalent. $$ (\exists x)(P_x) ...
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0answers
15 views

Disprove $\overline{\overline{A}} \leq \overline{\overline{B}}$ implies that $A \subseteq B$

Disprove $\overline{\overline{A}} \leq \overline{\overline{B}}$ implies that $A \subseteq B$ Counterexample: $A = \left \{1\right\}$ and $B = \left\{\varnothing\right\}$. $\overline{\overline{A}} = ...
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0answers
19 views

Prove that $(-\infty, b)$ for any $b \in \mathbb{R}$ has cardinality $c$

Prove that $(-\infty, b)$ for any $b \in \mathbb{R}$ has cardinality $c$ Proof: Define function $f: (-\infty, b) \rightarrow (-b, \infty)$ by $f(x) = -x$. Note that $f$ is one-to-one and onto because ...
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1answer
99 views

A polynomial is called a Fermat's polynomial…

A polynomial is called a Fermat polynomial if it can be written as the sum of the squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0) = ...
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1answer
41 views

Prove if $a$ and $b$ are real numbers, and $a \neq b$ and $a > 0$, $b > 0$, then $\frac{(a+b)}{2} > \sqrt{ab}$

Prove if $a$ and $b$ are real numbers, and $a \neq b$ and $a > 0$, $b > 0$, then $\frac{(a+b)}{2} > \sqrt{ab}$ Using a backward proof: $\frac{(a+b)}{2} > \sqrt{ab}$ $\Rightarrow ...
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3answers
143 views

Still struggling with proofs. [closed]

How do you construct rigorous math proofs on your own? Also how do you verify? I am finishing up my first semester of undergraduate analysis and still am struggling with writing proofs. Even though I ...
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1answer
46 views

Prove or disprove: If $B - A$ is nonempty, then $\overline{\overline{A}} < \overline{\overline{A \cup B}}$

Prove or disprove: If $B - A$ is nonempty, then $\overline{\overline{A}} < \overline{\overline{A \cup B}}$ Disprove (backwards proof): If $\overline{\overline{A}} < \overline{\overline{A \cup ...
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0answers
15 views

Prove or disprove: $\overline{\overline{A}} \leq \overline{\overline{B}}$ implies that $A \subseteq B$

Prove or disprove: $\overline{\overline{A}} \leq \overline{\overline{B}}$ implies that $A \subseteq B$ Disprove: Let $A = \left \{ 1 \right \}, B = \left \{ \varnothing \right \}$, then ...
0
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1answer
32 views

Fields and irreducible polynomial of $p^n$ degree

Let $K$ be a field of $p$ elements. Let $f(x) \in K [x]$ be an irreducible polynomial of degree $n$. Prove that the field $K[x]/(f(x))$ has $p^n$ elements. By given theorem, let $K$ be a field, ...
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1answer
73 views

What ring is the quotient $\mathbb{Z}[\sqrt{-11}]/(3,1+\sqrt{-11})$ isomorphic to?

Could anyone help me with this question? I've the feeling that the answer is $\mathbb{Z}/3\mathbb{Z}$, but I'm not sure at all and above all I don't know how to prove it. Thanks
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2answers
91 views

prove that $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$

I need help with proving this: $f:X\rightarrow Y$ is surjective if and only if $f(f^{-1}(C))=C$ $C\subseteq Y$ Thanks.
2
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1answer
54 views

Prove c exists for f:[0,1] -> R

Suppose that a function $f:[0,1] \to \mathbb R$ is continuous in $[0,1]$. Prove that there is a point $c\in [0,1]$ such that $\int_0^1 x^2 f(x)\,dx = \frac{f(c)}{3}$. Do I somehow use the mean value ...
2
votes
2answers
82 views

For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$

I don't understand the proof to the a/m claim. How we know that $\eta < \kappa$ and $(\alpha,\beta)<_{cw} (0, \eta) $ and hence $h: \mu \to \eta \times \eta$ is injective? Appreciate if anyone ...