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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3
votes
1answer
26 views

Prove: R∩R−1 is symmetric.

The problem that I'm having is proving it - obviously. The only context that I am provided with is: "Prove: R∩R−1 is symmetric." If (x,y) ∈ R then (y,x) ∈ R−1, and since it's the intersection, ...
0
votes
1answer
34 views

Prove that there is a surjection $ f:X \to Y$ if and only if $ |Y| \le |X| $.

Here's the problem: Let $X$ and $Y$ be sets. Prove that there is a surjection $$ f:X \to Y$$ if and only if $$ |Y| \le |X| $$. My work so far: I am working on the following direction: If $ |Y| ...
0
votes
0answers
24 views

combinatorial proof of Vandermonde's Identity [duplicate]

So I can not figure out the combinatorial proof for Vandermonde's Identity for the example $\sum_{i=0}^k \binom {k} {i}^2 = \binom {2k} {k}$ Any help would be appreciated. Figured it out, thanks :)
2
votes
4answers
73 views

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $ 7\mid b$

Prove if $7\mid a^2+b^2 \longrightarrow 7\mid a$ and $7\mid b$ What I did: I found the possible remainders for $a^2$ are $0, 1, 2$ and $4$. I think I should say $r_7(a^2)+r_7(b^2)$ can't equal any ...
1
vote
1answer
43 views

Prove that S is an equivalence relation on P(X)

Let $X = \{ 1, 2, 3, \ldots , 2015\}$ and $Y = \{ 1, 2, 3,\ldots, 271 \}$. Let S be the relation on power set (X) defined by For all A, B in $\mathcal{P} (X)$, $(A, B) \in S$ if and only if $|A \cap ...
0
votes
1answer
25 views

generalised eigenspace is invariant proof

let $T \in \text{End}(V)$ where $V$ is a finite dimension vector space. We define $V_j(\lambda) = \ker ((\lambda I - T)^j))$ as the generalised eigenspace. I am trying to prove that this space of is ...
2
votes
1answer
42 views

Every vector space has a basis

Prove that every vector space has a basis. I am going to use Zorn's lemma for this also here is a necessary definition regarding totally ordered subsets: one element will be contained in the other. ...
3
votes
5answers
146 views

Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$

The problem i have is: Prove that $2^n\le n!$ for all $n \in \mathbb{N},n\ge4$ Ive been trying to use different examples of similar problems like at: ...
0
votes
2answers
22 views

Short proofs about integrability

If true, the prove it; if false, the provide a counterexample. a) If $f$ is integrable, but $g$ isn't, then $f + g$ is not integrable. True: Assume that $f + g$ is integrable, then $f$ and $g$ must ...
0
votes
1answer
50 views

Probability proof and graphs

Let $G = (V, E)$ be a graph with vertex set $V$ and edge set $E$. A subset $I$ of $V$ is called an independent set if for any two distinct vertices $u$ and $v$ in $I$, $(u, v)$ is not an edge in $E$. ...
1
vote
2answers
45 views

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$

If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$ "$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss ...
0
votes
0answers
90 views

Edge and Vertex set proof using an algorithm

Disclaimer: This is a homework question, so no direct answers please. All that I'm looking for is a good springboard to get started from with this question, as it has been tearing me apart for the ...
0
votes
0answers
35 views

Existence of non-coprime between an integer and an arithmetic sequence

Take two relatively prime numbers $m,n \in \mathbb{Z}$ (i.e. $gcd(m,n) = 1$) where $m \neq 1$. Show that: $$\forall a \in \mathbb{Z} \textrm{ s.t. } 0<a<n$$ $$\exists i \in \mathbb{Z^+} ...
-1
votes
5answers
85 views

Prove that the sequence $\sin\left(\frac{n\pi}{3}\right)$ diverges

I don't want to hear that since $sin$ is a periodic function, etc, then we are done. I would like to see a simple proof that make use of the definition of convergence of a sequence. I have tried to ...
0
votes
1answer
29 views

Examples on how to give a proof or a counterexample of a statement

Examples; Prove or give a counterexample of the following statements,with quantifiers: 1) For each non-negative number s, there exists a non-negative number t such that s≥t 2) For each non-negative ...
1
vote
2answers
30 views

how to prove the uniqueness and existence of equations

I've the equation $e^x=5$, know it has the solution $x=\ln 5$. How to prove the existence before, and after the uniqueness of this solution?
1
vote
3answers
17 views

Prove $\sup S \leq \inf T$, if $s \leq t$, $\forall s \in S$ and $\forall t \in T$

I have the following exercise: Prove $\sup S \leq \inf T$, if $s \leq t$, forall $s \in S$ and $t \in T$. Note that $S$ is bounded above and $T$ is bounded below. This might seem too obvious, ...
0
votes
2answers
52 views

If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$

Just wanted to confirm that this is a correct solution: Proof: Suppose $f(x_0) > 0$ for some $x_0 \in [a,b]$. Then, by continuity of $f$, for $\epsilon < f(x_0)$, there exists $\delta > 0$ ...
4
votes
5answers
273 views

Show that $g(a) = g(b) = 0,\ \int_a^b f(x)g(x)dx=0 $ implies $f(x)=0$

Suppose $f$ is continuous on $[a, b]$, if for every continuous function $g$ on $[a, b]$ with $g(a) = g(b) = 0, \int_{a}^{b}f(x)g(x) dx = 0$, Show $f(x) = 0, \forall x \in [a, b]$, I want to ...
2
votes
2answers
51 views

Lebesgue outer Measure of a face of rectangle in $\Bbb R^{n}$

Show that the outer measure of a face $I_1 \times \dots \times I_{i-1} \times \{a\} \times I_{i+1} \times \dots \times I_n$ of a rectangle $I_1 \times \dots \times I_n \subset \Bbb R^{n}$ is zero. ...
1
vote
4answers
62 views

Prove $(2n + 1) + (2n + 3) + \cdots + (4n - 1) = 3n^2$ by induction

This might be an easy problem for you, but I am having difficulties in understanding the formula. As we can see, we have a pattern $$2n + \text{odd number}$$ in $$(2n + 1) + (2n + 3) + \cdots + ...
0
votes
2answers
46 views

Do I have the right start for this proof?

I'm trying to prove the following, Suppose R is a partial order on $A$, $B\subseteq A$, and $b\in B$. Prove that if $b$ is the smallest element of $B$, then it is also the greatest lower ...
1
vote
1answer
69 views

Bounding the edges belonging to no perfect matching

We are told to let $G = (X \cup Y, E)$ be a bipartite graph with $|X|=|Y|=n$, and to suppose that $G$ has a perfect matching. I am trying to find a way to prove that $G$ has at most $n \choose 2$ ...
-1
votes
2answers
37 views

Proof of factor.

If $m$ and $n$ are two positive integers, prove that $x+5$ is the factor for $(x+b)^{n} + (x+4)^{2m+1}$ How to write a proof on this ?
1
vote
3answers
47 views

Prove by induction $n^2 \leq n!$ for $n\geq 4$.

I managed to get $P(4):4^2 = 16 \geq 24 = 4!$ But then assuming $n^2 \geq n!, \forall n\geq4\in\mathbb{Z}$, I need to prove $(n+1)^2 \geq (n+1)!$ I tried $n^2+2n+1\geq n!\cdot (n+1)$, but I got ...
2
votes
2answers
76 views

T/F: $\forall \epsilon , \exists \delta \gt 0$ s.t. $\left| f(x)-f(a) \right| \lt \epsilon \implies \left| x-a \right| \lt \delta $

Here's the question: Is the following true or false? There is a function $f: \mathbb R \to \mathbb R$ that satisfies the following condition: For every $a \in \mathbb R $ and $ \epsilon \gt 0 $ ...
0
votes
1answer
42 views

Proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous

Could you tell me some webpages or books where I can find the proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous. I know how it can proof if $P:\mathbb{R}\to\mathbb{R}$, but I don't know ...
1
vote
1answer
40 views

Use the principle of mathematical induction to prove that $n \lt (\frac{3}{2})^n$ for all integers $n \ge 1$.

Here's the problem: Use the principle of mathematical induction to prove that $n \lt (\frac{3}{2})^n$ for all integers $n \ge 1$. Here's what I've got: Base Case: $1 \lt (\frac{3}{2})^1$ is true. ...
3
votes
2answers
57 views

Continuous function with finitely many discontinuities is Riemann Integral

After a lecture today, I just wanted to confirm that I understand the proof of the following: If $f: [a,b] \to \mathbb{R}$ is bounded and continuous and has finitely many discontinuities, $f \in ...
0
votes
2answers
55 views

A question on the proof of commutativity of the sum of natural numbers?

I made this question yesterday and today I've been thinking about another aspect of it. But this question is totally related to the previous one: I am trying to make a clarification about a proof of ...
0
votes
2answers
45 views

Multidimensional Proof by Induction

I have been given a recursive relation $$f(m,n)=f(m−1,n)+f(m,n−1)$$ in which I need to prove by mathematical induction that, $$f(m, n) = {(m + n)!\over(m!n!)}$$ over all natural numbers where $$f(0, ...
3
votes
0answers
44 views

In a math paper, what is a remark?

I sometimes see paragraphs labeled 'Remark.' However, papers that include remarks also include unlabeled explanatory paragraphs (i.e. all the other writing in the article) that seem to be remarks. ...
1
vote
4answers
486 views

Suppose that x and y are irrational, but x + y is rational. Prove that x - y is irrational. [closed]

I can understand how it works in my head, I don't know how to prove it though.
0
votes
1answer
32 views

Boolean algebra proof - I don't know why this is valid!

So this is the answer proof I was given, I'm stumped by the final application of the Idempotent law (where does that 1 come from!?) As I understood it a 0 or 1 can only come from a combination of A ...
0
votes
1answer
36 views

Show any open interval is a half open set?

How do I show that any open interval is an half open set and use this to conclude that any open set is also half open? I am in an introduction to proofs writing class. I have a feeling I need to use ...
3
votes
0answers
38 views

My proof that sum of convergent sequences converges to sum of limits

Does my proof appear correct? Also, do you like the notation? $\textbf{Theorem.}$ If $(a_n)_{n \in \mathbb{N}}$ and $(b_n)_{n \in \mathbb{N}}$ are convergent real sequences, then $$ \lim_{n \to ...
0
votes
1answer
19 views

Let X be a nonempty set. Let x∈X. Show that the collection 𝔗={U⊆X:U=∅ or x∈U} is a topology for X.

Let $X$ be a nonempty set. Let $x \in X$. Show that the collection $ \mathfrak T = \{ U \subseteq X : U = \emptyset$ or $ x \in U \}$ is a topology for X. I know I need to show that this ...
0
votes
2answers
30 views

What would the correct English description be for the difference for 1/3

If you measure a task & it takes 3 seconds, then the next time you do the same task, it takes you 1 second, is the difference 200% or 67%? Or would you say the difference is 200% because 3-1=2 ...
0
votes
1answer
44 views

Proof by contradiction or contrapositive sets help

so I'm having difficulties proving the following Theorem, through either proof by contradiction or contrapositive. Can someone please help me? The problem is as follows: Prove that for any two sets, ...
0
votes
1answer
47 views

Iterating proof step

Many books proves theorems by performing one proof step and using this step as a scheme they say by repeating this step $l$ times we prove that... I wonder whether there is some formal meta-theorem ...
3
votes
1answer
55 views

Prove that $n(r) < 2\pi \sqrt[3]{r^{2}}$

Suppose that $n(r)$ denotes the numbers of points with integer coordinates on a circle of radius $r > 1$. Prove that $$ n(r) < 2\pi \sqrt[3]{r^{2}} $$ What process would you use to resolve ...
1
vote
3answers
44 views

Mathematical Induction Proof Question dealing with integers

How would you use mathematical induction to prove that $1 \cdot 2 \cdot 3 + 2 \cdot 3 \cdot 4 + \dots + n \cdot (n + 1) \cdot (n + 2) = \frac{n(n + 1)(n + 2)(n + 3)}{4}$ I tried proving the base ...
1
vote
2answers
75 views

How to prove that the $\lceil x y \rceil \le \lceil x\rceil\lceil y\rceil$ for real numbers $x, y$?

I know intuitively that this is true, but whenever I try and break apart that intuition to see where it's coming from, I essentially end up re-writing the assumption I'm trying to prove. I've tried ...
0
votes
1answer
48 views

Proof that minimum of sum of absolute differences is greater or equal of max value minus min value

Let's have an vector of natural numbers $[v_1, ..., v_N]$ my goal is to show that $$\sum_{i=1}^{N-1}|v_i - v_{i+1}| \ge v_{max} - v_{min}$$ where $v_{max} = \max_{i\in1...N}(v_i)$ and $v_{min} = ...
2
votes
1answer
69 views

Prove by induction that numbers of Fibonacci of the form $F_{3n}$ are even

I am not quite asking to solve this problem, but I am asking what they are asking me. This is the problem: The Fibonacci numbers $F_n$ for $n \in \mathbb{N}$ are defined by $F_0 = 0$, $F_1 = 1$, ...
1
vote
3answers
79 views

How to resolve $x \in A \wedge x \notin A $?

Let A and B be two sets. Then $A \setminus B = \{x: x\in A \wedge x\notin B\}$ $A \setminus B = \{x: x\in A \wedge x\notin A \cap B\}$ How can one prove that two logical statements are equal? ...
0
votes
2answers
44 views

Computation of determinant for Using Inverse Function Theorem

Let $f : \Bbb R^{3} \setminus \{(0, 0, 0)\} → \Bbb R^{3} \setminus \{(0, 0, 0)\}$ be given by $f(x, y, z) = (x/(x^{2} + y^{2} + z^{2}), y/(x^{2} + y^{2} + z^{2}), z/(x^{2} + y^{2} + z^{2}))$. Show ...
1
vote
1answer
68 views

How do I prove that every open interval that contains $ \{ 1,2\}$ must also contain 1.5?

How do I prove that every open interval that contains $ \{ 1,2\}$ must also contain 1.5? $| x -x_0| \lt \varepsilon$ if and only if $x$ is in the interval $(x_o -\varepsilon, x_o + \varepsilon)$ We ...
0
votes
3answers
50 views

Prove that $| x -x_0| \lt \varepsilon$ if and only if $x$ is in the interval $(x_o -\varepsilon, x_o + \varepsilon)$.

Let $x_0$ and $x$ be real numbers and let $\varepsilon$ be a real number with $\varepsilon \gt0$. Prove that $| x -x_0| \lt \varepsilon$ if and only if $x$ is in the interval $(x_o -\varepsilon, ...
0
votes
2answers
60 views

How do I prove that the complement of the closed interval $[a,b]$ is an open set.

How do I prove that the complement of the closed interval $[a,b]$ is an open set. I have a theorem that says an open set is a union of open intervals. Can I simply say the complement of the closed ...