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.

learn more… | top users | synonyms

0
votes
2answers
79 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 ...
1
vote
1answer
39 views

Next step to take in direct proof for one to one?

This is from Discrete Mathematics and its Applications And the definition of strictly increasing. Here is my work so far. I know that a direct proof involves making an assumption p, which in ...
0
votes
0answers
23 views

How is Lagrange's $2\sqrt{D}$ bound on partial denominators proven for periodic regular continued fractions of quadratic irrationals

For the quadratic surd: $$ \zeta = \dfrac{P + \sqrt D}Q $$ the wikipedia article on periodic continued fractions mentions that Lagrange proves that the largest partial denominator of a regular ...
0
votes
1answer
98 views

How to prove that $\cos(\pi÷11)+\cos(3\pi÷11)+\cos(5\pi÷11)+\cos(7\pi÷11)+\cos(9\pi÷11)=0.5$? [duplicate]

I need to prove that $$\cos\dfrac{\pi}{11}+\cos\dfrac{3\pi}{11}+\cos\dfrac{5\pi}{11}+\cos\dfrac{7\pi}{11}+\cos\dfrac{9\pi}{11}=\dfrac{1}{2}$$ How to do it?
2
votes
4answers
79 views

Proving that the series 1 + … + $1 / \sqrt{x}$ < $2 \sqrt{x}$

Proving that the series 1 + ... + $1 / \sqrt{x}$ < $2 \sqrt{x}$ I am doing it by simple induction adding $1/\sqrt{x+1}$ to both sides, but I can't find a way to manipulate this expression and find ...
2
votes
2answers
94 views

Proving by induction that any two natural numbers are equal.

This is something I've been working on for a while now; although it seems trivial, I am confused. I can't seem to find the error. Originally I thought the problem was with the base case, then I ...
2
votes
1answer
40 views

Disprove that if $f([-2, 2])$ is an interval, then $f$ is continuous

Disprove that if $f([-2, 2])$ is an interval, then $f$ is continuous My counter example is $\begin{cases} 1 - x & \text{ if } -2 \leq x \leq 1 \\ 2 - x & \text{ if } 1 < x \leq 2 ...
2
votes
2answers
42 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
1
vote
1answer
18 views

Proof of a tree with a vertex of degree k and less than k vertices of degree 1

The question is : Does there exist a tree with a vertex of degree k and less than k vertices of degree 1? I tried a lot but it is impossible to find. There is no tree with a vertex degree k and less ...
0
votes
1answer
16 views

Finding a bijective function from $\prod_{i\in I}X_i$ to $\bigl(\prod_{j\in J}X_j\bigr)\times\bigl(\prod_{k\in K}X_k\bigr)$

If $(X_i)_{i\in I}$ is a family of sets and $J,K$ are non-empty disjoint sets of $I$ such that $I=J\cup K$, then show that there is a bijective function from $\prod_{i\in I}X_i$ to ...
0
votes
2answers
77 views

Proving a slight variation of the fibonacci formula using complete induction

I learned this formula for the Fibonacci series, and its respective proof in one of my Computer Science classes. F(0) = 0; F(1) = 1; F(2) = 1 However, I am taking an abstract mathematics class and ...
1
vote
4answers
112 views

Why can't I prove this statement by simple induction? Sum of $1/2^1 + \cdots+ n/2^n = x$

I have to prove the following: $$ \frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3}+\cdots+\frac{n}{2^n}=2-\frac{2 + n}{2^n}. $$ I am trying to prove this by simple induction. First, I proved that $P(1)$ ...
0
votes
1answer
36 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
0
votes
1answer
55 views

Let ${a_n},{b_n}>0 ,\lim \limits_{n \to \infty} [a_n+b_n]=0 $ then$ \lim \limits_{n \to \infty}a_n=0 $ and$ \lim \limits_{n \to \infty} b_n = 0$

Let ${a_n}$ and ${b_n}$ be sequences of nonnegative numbers. Show that if $\lim \limits_{n \to \infty} [a_n+b_n]=0$ then $\lim \limits_{n \to \infty}a_n=0$ and $\lim \limits_{n \to \infty} b_n = 0$. ...
0
votes
1answer
94 views

Property of continuous functions regarding maximum

Claim 1: If $f: [a, b] \rightarrow \mathbb{R}$ is continuous, then $f$ assumes a maximum value I know there's a theorem that states if $f$ is a continuous real-valued function on a closed interval ...
-1
votes
2answers
121 views

How to show that “Uniformly continuous implies continuous”? [closed]

Can I go from the definition of uniformly continuity to continuity? Please somebody show me how to do that. Thanks.
1
vote
2answers
100 views

How to find all the equivalence classes for a specific equivalence relation?

What are the equivalence classes of the following equivalence relation $$S=\{(x,y) \in \mathbb{R} \times \mathbb{R} \mid x - y \in \mathbb{Q} \}$$? I know that an equivalence relation $R$ on a set ...
0
votes
1answer
18 views

If there are $n$ $Y$'s for every $X$, but $m$ $X$'s for every $Y$, then the total number of $Y$'s is $n/m$ times the number of $X$'s?

(Apologies for the lengthy title. If you can make it shorter but still descriptive, please do so.) There is a certain form of reasoning that I find myself using every once in a while. I believe it's ...
0
votes
0answers
33 views

proving prime number's divisors

Let p ̸= 0, ±1 be an integer. Prove that p is prime if and only if p satisfies the following property: Whenever a and b are integers such that p = a · b, either a = ±1 or b = ±1. I proved the forward ...
1
vote
1answer
29 views

$S$ is a reflexive closure of $R$. Prove that if $R$ is symmetric, then so is $S$

Suppose $R$ is a relation on $A$, and let $S$ be the reflexive closure of $R$. Prove that if $R$ is symmetric, then so is $S$. S is the reflexive closure of $R$, which means that $$\forall x ...
0
votes
1answer
34 views

If $R_1$ and $R_2$ are symmetric and transitive, prove that also $R_1 \cup R_2$ is symmetric and transitive

I have to prove or confute (with a counter example) that, if $R_1$ and $R_2$ are symmetric and transitive, then $R_1 \cup R_2$ is also symmetric and transitive. Note: To prove the transitive case, ...
2
votes
2answers
119 views

If $n^2$ is even $n$ is even

I understand that there are already several answers to how to prove this question: Prove if $n^2$ is even, then $n$ is even. Prove that if $n^2$ is even then $n$ is even I am trying to understand ...
2
votes
1answer
50 views

Next step to reach the contradiction?

This is a problem from Discrete Mathematics and its Applications Here are my notes and my current work so far for this problem. I started with an assumption that what i am trying to prove is ...
0
votes
2answers
59 views

What to use for r in proof by contradiction?

This is a problem from Discrete Mathematics and its applications To this proof, I am trying to use proof by contradiction. Here is how the book described the process of proof by contradiction. I ...
0
votes
1answer
50 views

Next step to take in direct proof?

This is a problem from Discrete Mathematics and its Applications. I understand the basic ideas of the direct proof. Basically a proof is a conclusion from a series of steps to establish the truth of ...
2
votes
1answer
50 views

Next step to take in direct proof or a workaround around current dilemma?

This is a problem from Discrete Math and Its Applications I used a direct proof to do this proof. I understand the process/idea behind the direct proof, mainly (from ...
2
votes
1answer
66 views

$\exists x (P(x) \to \forall y P(y))$ [duplicate]

Prove $\exists x (P(x) \to \forall y P(y))$. Let x = y. Suppose P(x) is true. Let y be arbitrary. Since P(x) is true, it must be that P(y) is true. Since y was arbitrary, we can conclude that ...
1
vote
2answers
113 views

How to calculate $\sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$ [closed]

How can I calculate the summation: $\displaystyle \sum\limits_{n=-\infty}^\infty \frac{1}{1+n^2}$
0
votes
1answer
31 views

Cases for x in $ \forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x) $.

This is from Velleman p145, problem 28. Theorem: $\forall x \in \mathbb{R} \exists y \in \mathbb{R} (xy^2 \neq y - x)$. Author's Proof: Let x be an arbitrary real number. Case 1. $x = 0$. Let $y ...
0
votes
2answers
18 views

If $g\circ f$ is $1$-$1$ then $f$ is $1$-$1$ but $g$ is not necessarily $1$-$1$.

Let $f:X\longrightarrow Y$ and $g: Y\longrightarrow Z$. Show that, if $g\circ f$ is $1$-$1$, then $f$ is $1$-$1$, but $g$ is not necessarily $1$-$1$ I don't know how to start the proof. We have ...
2
votes
2answers
107 views

Must proofs always be cited (Thesis)?

I have some proofs of theorems in my thesis that are very similar to the proofs from the literature ( "my" proofs are more extended and have more explaination, the structure isn't the same either). ...
1
vote
2answers
101 views

Proof of $\exists x(P(x) \Rightarrow \forall y P(y))$

Exercise 31 of chapter 3.5 in How To Prove It by Velleman is proving this statement: $\exists x(P(x) \Rightarrow \forall y P(y))$. (Note: The proof shouldn't be formal, but in the "usual" ...
1
vote
1answer
74 views

Prove that a function is continuous at $x = x_{0}$ using the $\delta - \epsilon$ definition

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$ $\forall \epsilon > 0$, $\exists \delta = ?$ ...
0
votes
1answer
67 views

Prove that function $f$ is continuous at $x = x_{0}$

In class we're given the following definition about continuity, and I want to apply this definition to the problems that follow: $f$ is continuous at $x_{0} \in \mathrm{dom}(f)$ if $\forall x_{n} \in ...
2
votes
1answer
76 views

Prove that $f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$ is continuous at $0$

Prove that $$f(x) = \begin{cases} x^2 & \text{ if } x\in \mathbb{Q} \\ 0 & \text{ if } x\in \mathbb{Q}^{c} \end{cases}$$ is continuous at $0$ and discontinuous everywhere else Proof: ...
0
votes
2answers
17 views

Why do I need to know that rng R = A for this proof?

Let $A$ be a nonempty set. Show that if $R$ is a symmetric and transitive relation on $A$ such that $rngR = A$, then $R$ is reflexive on $A$. So I proved this by saying: For all $x,y\in A$, $(x,y)\in ...
6
votes
2answers
104 views

How can I prove whether a $9\times 9$ square can be filled with L-shaped pieces in a completely “regular” way?

There are a great many ways to fill a $9\times 9$ square with L-shaped pieces. One of them is below. Now, note that there are eleven $2\times 3$ rectangles that are formed, as well as a larger L ...
0
votes
1answer
71 views

$f(A\cap B)=f(A)\cap f(B)$. Where's the mistake?

I'm trying to prove something that is false, to see where is the contradiction. I want to prove that if $f:X\longrightarrow Y$ and $A,B\subseteq X$ then $f(A\cap B)=f(A)\cap f(B)$. So, let $y \in ...
1
vote
4answers
39 views

Proving the remainder when a polynomial is divided by an integer.

How should I go around proving that $\forall x \in \mathbb{Z}$, the remainder when $x^2+2x$ is divided by $3$ is $0$ or $2$? Do I use the division algorithm for this one?
0
votes
2answers
20 views

Proving using the definitions of “strictly dominated by” and “dominated by”

Let $A, B,$ and $C$ be sets. If $A$ is strictly dominated by $B$ and $B$ is dominated by $C$, then $A$ is strictly dominated by $C$. I need to prove this using the definitions of "dominated by" ...
1
vote
2answers
54 views

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous

Prove that $f(r) = 0$ for $\forall r \in \mathbb{Q} \Rightarrow f(x) = 0$ for $\forall x \in \mathbb{R}$ given $f$ is continuous. I know that the Dirichlet function is discontinuous everywhere ...
0
votes
0answers
58 views

Proof using Archimedean property and Bernoulli's inequality

I am trying to prove the theorem below (using both the Archimedean property and Bernoulli's inequality). As usual, I would like to write a highly intelligible proof. Any constructive feedback is ...
0
votes
1answer
47 views

A proof that the Cantor set is Perfect

I found in a book a proof that the Cantor Set $\Delta$ is perfect, however I would like to know if "my proof" does the job in the same way. Theorem: The Cantor Set $\Delta$ is perfect. ...
3
votes
4answers
82 views

Proof of divisibility using modular arithmetic: $5\mid 6^n - 5n + 4$

Prove that: $$6^n - 5n + 4 \space \text{is divisible by 5 for} \space n\ge1$$ Using Modular arithmetic. Please do not refer to other SE questions, there was one already posted but it was using ...
-1
votes
1answer
38 views

Proof that this set is convex set

I need a help with prooving that a given set is a convex set: $A \in \mathbb{R}^{m\times n}, c \in \mathbb{R}^n , b\in \mathbb{R}^m:$ $? \in argmin\{ c^Tx| Ax=b,x \geq 0\}$ I know the definition of ...
0
votes
1answer
46 views

$xy \le xz$ if both $y \le z$ and $0 \le x$. (very easy proof exercise)

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem Let $x,y,z \in ...
3
votes
3answers
45 views

If $f$ continuous in $[-1,1]$ Then $g(x)= \int_{-x}^x \! \, f(t)dt$ derivative in $[-1,1]$ and $g'(x)=f(x)+f(-x)$

I have this problem its a proof/disproof problem. For some reason I get wrong answer. If $f$ continuous in $[-1,1]$ Then $g(x)= \int_{-x}^x \! \, f(t)dt$ derivative in $[-1,1]$ and ...
3
votes
1answer
115 views

$x^n y^n = (xy)^n$, proof exercise

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem \begin{equation*} x^n ...
0
votes
0answers
31 views

Proving the solutions to an Equation

Consider: I am sort of confused. Without using the intermediate value theorem, directly. I suppose indirect use if fine. $1 + x^2 + \sin^2(x) > 0$ for $x \in \mathbb{R}$ Let the $LHS = I$ $$I ...
0
votes
3answers
45 views

Is this a valid sum formula for rational functions?

Consider: $$\frac{a(x)}{\displaystyle \sum_{n=1}^{\infty} (-1)^{n-1}x^{2n-2}}$$ Is the expressions above equivalent to: $$\sum_{n=1}^{\infty} \frac{a(x)}{(-1)^{n-1}x^{2n-2}}$$ ??