For questions about the formulation of a proof, not about the mathematics behind it.

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1answer
28 views

How to prove that bijective functions have surjective inverses

I am trying to prove Since $f$ is bijective therefore it has an inverse and $f^{-1}$ is bijective But stuck on proving that $f^{-1}$ is surjective Suppose $f: A \to B$ is a bijection, then a ...
2
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1answer
27 views

Simple Graph Degree Sequence Proof

Let G be a simple graph with degree sequence $(d_1,d_2,...,d_n)$. Prove that for each k, $0<k<n$: $$\sum_{i=1}^k d_i\le k(k-1)+\sum_{i=k+1}^n min(k,d_i)$$ I'm new to graph theory and proof ...
2
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2answers
36 views

Can I use “$\iff$” symbol when I “transform” an expression to another form?

I am writing a solution to prove that $\sqrt5$ is not rational. Here is my first half proof: Assume $\sqrt{5}$ is a rational number. By the definition of rational number, $\sqrt{5} = \frac{p}{q}$, ...
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0answers
66 views

Algorithms in formal logic + ZFC

From my understanding, most of mathematics can be built up assuming a mechanical procedure of manipulating finite strings of symbols according to certain rules. A conventional way to do it is via the ...
0
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1answer
34 views

Simple graph proving cardinality of a set is even

Let G be a simple graph, and let S be the set of vertices of even degree, and T be the set of vertices of odd degree. Prove that the cardinality (number of things in) T is even. I am new to graph ...
2
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2answers
36 views

Proof that if $S$ is closed then $S^c$ is open from Pugh's text

Recall $S$ is closed if it contains all the limit points and $S$ is open if the metric $d(p,q) < r$ around every $q \in S$ for some $r>0$ Pugh's real analysis provides a proof as follows: ...
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1answer
21 views

What is the biggest computation one should use for an induction base

First of all sorry for me (informal) language here - i am not a native english speaker. I have tried to look up my question here but couldnt find anything. I also tried to look it up on google but i ...
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1answer
84 views

Any advice to help visualizing/understanding theorems in Real Analysis?

I am currently taking a real analysis course and sometimes I can't really get the complete meaning/details of certain theorems. For example, one of the statements in the Heine-Borel Theorem saying ...
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2answers
34 views

Proving sum of product forms a pattern in n * nnnnnn…

I am consider a problem regarding numbers which are, in decimal, one digit repeated - for instance, $88888888$ is such a number. In particular, I am looking at the following problem: The sum of ...
1
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1answer
27 views

Proving the additive identity and additive inverse of a set

Let V denote the set of ordered pairs of real numbers. For $(a_1, a_2), (b_1,b_2) \in V, (a_1, a_2)+(b_1,b_2)=(a_1+b_1, a_2b_2)$. I am trying to prove that a set V is (or is not) a vector space and I ...
2
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1answer
91 views

Prove $\int_0^{\pi/2}{(\sin^nx-\cos^nx)}dx=0$

How do I show that $$\int_0^{\pi/2}{(\sin^nx-\cos^nx)}dx=0$$ for all $n>0,n\in Z$ I know that it is true, but I don’t know how to prove.
2
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1answer
35 views

Trouble with Discrete Math proof

This seemingly easy proof is giving me some trouble.. For every number n $\in$ $Z$, if $n>n^2+1$ then $n\leq0$. I find that proving the conditional statement P implies Q is false. For example ...
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2answers
42 views

Any good resources/reference books about bijective proofs (Combinatorics)?

I am currently taking a Combinatorics course in this sem,however, my prof hasn't talked much about how to construct a bijection between two sets we want to count. I have searched online but I still ...
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2answers
53 views

If$(10000x^{2015} - x)\over (x^2+x+1)$is odd, then $4x^2 + 3x + 1$ is even.

How would I I begin to prove this implication? Starting with the hypothesis $(10000x^{2015} - x)\over(x^2+x+1)$ = $2(k)+1$. I'm sort of lost on where to go?
2
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0answers
33 views

Proof of hockey stick like theorem for multinomial coefficients

I discovered following identity via a programming exercise. Looking for the way how it prove it by induction or some other way. $$\binom{a_1+a_2+\cdots+a_t}{a_1,a_2,\cdots,a_t}=1+\sum_{i=2}^t ...
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2answers
83 views

What is a combinatorial proof exactly?

It almost seems as though a combinatorial proof is "Explain the intuition behind this relationship (using normal words) to explain why it is true." I'm a little lost as to how this is a proof ...
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0answers
37 views

Fubini's theorem and set of measure zero

Let A be a rectangle in $R^k$ , let B be a rectangle in $R^n$, let Q = A x B. Let f: Q $\rightarrow R$ be a bounded function. Show that if $\int_Q {f}$ exists, then $\int_{y \in B} {f(x,y)} $ (denoted ...
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4answers
65 views

Prove if $2|x^{2} - 1$ then $8|x^{2} - 1$

I have seen this question posted before but my question is in the way I proved it. My books tells us to recall we have proven if $2|x^{2} - 1$ then $4|x^{2} - 1$ Using this and the fact $x^{2} - 1 ...
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0answers
22 views

Jordan Measure and Riemann Integration

Let f,g : S $\rightarrow$ R; assume f and g are integrable over S. Show that if f and g agree except on a set of measure zero then $\int_s {f}$ = $\int_s {g}$. Attempt at a solution: Choose a ...
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0answers
17 views

Equivalence relation on $S^2$ and unit square make same space

I think I understand a bit of this task, but I hope someone can look critical to my answers: Let $X$ be the space obtained from the sphere $S^2$ by gluing the north and the south pole (with the ...
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2answers
61 views

How to prove ceiling and floor inequality more 'formally'?

The inequality in question is below: $x - 1 < \lfloor x\rfloor \le x \le \lceil x \rceil < x + 1 $ Essentially, I must prove the above for every real number $x$. To begin this proof, I broke ...
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1answer
16 views

Prove this proposition by cases

When i try to prove this proposition by cases $$x^2 + w^2 + y^2 = z^2$$ Where $x$, $y$, $w$ are positive integers .. $z$ is even if and only if $x$, $y$ and $w$ are even I represent even as $2i$ ...
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1answer
58 views

Proving every infinite set $S$ contains a denumerable subset

I have some trouble understanding this proof of every infinite set $S$ contains a denumerable subset contained in Charles Pugh's Real Analysis text. Proof: Since $S$ is infinite, it is ...
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2answers
56 views

Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $ [closed]

Let $a_1, a_2, a_3,\ldots a_n$ be positive real numbers where $n > 1$. Prove that $$1-\displaystyle \sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $$ Can this be proved using the binomial theorem ...
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1answer
26 views

Bijection for Rook placecement and Stirling number of 2nd kind

Say we have an nxn chessboard from which the squares below the diagonal are removed to obtain a new board $C_n$. The board $C_3$ is shown below. Let the number of ways to place k non-attacking ...
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3answers
44 views

Find the solution of the recurrence relation (fibonacci)

Find the solution of the recurrence relation $f_n = f_{n-1} + f_{n-2}$ with $f_0 = f_1 = 1$ I had someone show me how to solve/prove this using the linear difference equation, but I had never been ...
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0answers
34 views

Quick way to describe a 'top 10' subset

I have a finite ordered set $\mathcal{X}$ and some weight $f:\mathcal{X}\rightarrow \mathbb{R}$ and I wish to quickly specify a subset of it $S_{N}(a)$ which is "The $N$ elements of $\mathcal{X}$ ...
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1answer
32 views

Proving There Exists a Quadratic Function that Shares Tangent Lines With Another Quadratic

I have been working on proving a theorem and I think I have it, I'm still new to proofs so any advice would be greatly appreciated! Theorem: Let $f(x) = Ax^{2}+Bx+C_{1}$. Let $y_{1}(x) = f'(Q)x+b$ ...
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0answers
31 views

Prove $\int_0^x f^3 \le (\int_0^x f)^2$ [duplicate]

$f$ is a differentiable function such that $f(0)=0 \text{ and } 0<f’\le 1$. We need to prove that for every $x\ge 0$, $\int_0^x f^3 \le (\int_0^x f)^2$. How do we show this?
3
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1answer
106 views

Prove that $(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$

For the non-negative real numbers $a, b, c$ prove that $$(a^2+2)(b^2+2)(c^2+2)\geq 3(a+b+c)^2$$ What I did is applying Holder's inequality in ...
2
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1answer
31 views

Conjecture for the maximum number of rooms.

The puzzle I have is essentially this, but for $n$ rows.For this instance of $n=5$, quick tallying reveals the answer to be $21$. For $n=4$, it is $13$. For $n=3$, it is $7$. For $n=2$ it is $3$. ...
0
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1answer
24 views

Writing the needed or the known as clarification?

Suppose $\kappa > 2 \ $ and $a,b \ $ two positive real numbers satisfying $a > b$. If somewhere in a proof about $a, b \ $ I would need the inequality $a^{-\kappa} < b^{-\kappa}$, how should ...
1
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1answer
49 views

Critique of my proof of the structure of ideals in a quotient ring

I am wondering if there are slicker ways to prove this: If let $A$ be a commutative ring and $I \subseteq A$ be an ideal, and let $\nu : A \rightarrow \frac{A}{I}$ be the quotient homomorphism. Then ...
3
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0answers
52 views

Proof for $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} + \frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} + \frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$? [duplicate]

Let $a, b, c$ be real numbers such that $abc=8$. Prove that: $$\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} + \frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} + \frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$$ ...
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0answers
26 views

Average Sum bigger than average product?

How can I prove that $\sum_{k=1}^K p_kt_k > \prod_{k=1}^K t_k^{p_k}$ given that $t_k>0$, $0<p_k<1$ and $\sum p_k = 1$ Thank you
2
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1answer
120 views

Show that $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$

For positive real numbers $a,b,c$ with $abc = 8$ prove that $$ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} + \frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} + \frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}. $$ Can ...
0
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1answer
21 views

Matrix multiplication identity proof

How can I prove that $(PQ + I_N)^{-1}P = P(QP + I_M)^{-1}$ knowing that we have two matrix $P_{N \times M}$ and $Q_{M \times N}$. Thank you very much for help.
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2answers
34 views

Showing an inequality is true using a counterexample of the opposite direction

This semester, I took a course in real analysis and my proof skills were mediocre at best. Unfortunately, I took to Google for many answers since I could not "start" proofs in the slightest. A ...
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0answers
30 views

Real analysis supremum proof

Let A be a non-empty bounded sub-set of $\mathbb{R}$. Let $B\subset\mathbb{R}$, given by:$B=\{\frac{a_1+2a_2}{2}|a_1,a_2\in A\}$. Express $\sup B$ in terms of $\sup A$. My attempt: Suppose ...
1
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1answer
33 views

How can you prove the inequality $2x^3 (x^3 + 8y^3) + 2y^3 (y^3 + 8z^3) + 2z^3 (z^3 + 8x^3) ≥ 9x^4 (y^2 + z^2) + 9y^4 (z^2 + x^2) + 9z^4 (x^2 + y^2)$

I changed the RHS as $\displaystyle \sum_{cyc} 9x^4(S-x^2) $ for $S = x^2 + y^2 + z^2$ Then I thought I could apply Jensen's inequality for $f(x) = 9x^4(S-x^2) = -9x^6 + 9Sx^4$ in the RHS as follows ...
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4answers
1k views

Is collapsing considered a legitimate proof?

For example if I want to prove that $2^n - 1 = 1 + 2 + 4 + 8 +...+ 2^{n-1}$ I can obviously use induction and that is accepted. But I can also collapse it like: To Prove $2^n = S(n)$: $S(n) = (1 + ...
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1answer
31 views

Pick any three digit number

A Link to it is here: http://qr.ae/Rgrdrk In short, you take a 3 digit number, subtract the sum of it's digits. Any given digit in the result is equal to the difference between the other two numbers ...
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2answers
60 views

If $A$ an $B$ are finite, the the set of all functions from $A$ to $B$ is finite..

Suppose $A$ and $B$ are finite $\rightarrow$ $A \approx N_k$ $\land B \approx N_m$ then $A = (x_1,...,x_k)$ $\land B = (y_1, ..., y_k)$ Let $L$ be the set of all functions $A \rightarrow B$ , ...
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2answers
29 views

Existence of maximum and minimum

Let $f:\mathbb{R}_+\rightarrow \mathbb{R}$ be continuous and such that $f(0)=1$ and $lim_{x\rightarrow+\infty}f(x) = 0$. Prove that $f$ must have a maximum in $\mathbb{R}_+$. What about the ...
4
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2answers
159 views

Center of the Quaternions: Proof and Method

I have to calculate the center of the real quaternions, $\mathbb{H}$. So, I assumed two real quaternions, $q_n=a_n+b_ni+c_nj+d_nk$ and computed their products. I assume since we are dealing with ...
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0answers
10 views

Choosing nonzero entries from an array so no pair in same row or column

Suppose we have an $n\times n$ array $A$ of non-negative real numbers in which the sum of each row and each column is $1$. We want to find $n$ entries of the array $(x_1,y_1), \dots, ...
1
vote
1answer
79 views

The set of invertible elements of a monoid is closed under multiplication [duplicate]

Let $M$ be a monoid and let $U(M)$ be the set of invertible elements of $M$. How can I prove that $U(M)$ is closed under the binary operation on $M$, i.e., that that $a \in U(M)$ and $b \in U(M)$ ...
1
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2answers
60 views

$f$ ia continuously differentiable function with $f'(c)=0$…

Let $f$ be a continuously differentiable function on $[a,b].$ There is a number $c$ in $(a,b]$ such that $f'(c)=0.$ Then prove that there is a fixed number $\xi\in(a,b)$ such that ...
0
votes
1answer
32 views

Proof of Binomial Coefficients Comparison Inequality

Please help to prove the inequality $$ \binom{a}{b}\leq\binom{a+j}{b+i}$$ For $i\leq j$ Using the basic identity $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ I have proceeded to $$ 1 \leq ...
3
votes
2answers
74 views

Proof that for any $x$ there is a $y$ such that $xy$ is a palindrome

I'm wondering how I would prove For any $x$ there exists at least one $y$ such that $xy$ is a palindrome. For example: 91*99=9009