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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2
votes
2answers
66 views

Prove or refute contingent: If A implies B is contingent, then B is too

The question is: If $A, A \to B$ are contingent, then so is $B$ $A, A \to B$ (implies) is a contingent, but how exactly to show «so is $B$»? If I'm using a truth table, how should I show that ...
1
vote
5answers
148 views

Given that $\alpha$ and $\beta$ are acute angles, show that $\alpha+\beta = \pi/2$ if and only if $\cos^2{\alpha} +\cos^2{\beta} = 1$.

This question is from an exam paper: Given that $\alpha$ and $\beta$ are acute angles, show that $\alpha+\beta = \pi/2$ if and only if $\cos^2{\alpha} +\cos^2{\beta} = 1$. I want to do it in ...
1
vote
1answer
42 views

If $S_n=\left[-\frac{n}{n+1},\frac{n}{n+1}\right]$ then $\bigcup\limits_{n\geq1}S_n=(-1,1)$

I was self reading Mathmatics for Economists by Simon and Blume. Consider the closed sets $S_n=\left[-\frac{n}{n+1},\frac{n}{n+1}\right]$ for $n\geq1,n\in\mathbb N$. Then ...
3
votes
1answer
147 views

Prove that there does not exist a surjective function from the set of rationals to reals.

Prove that there does not exist a surjective function f: $\mathbb{Q}\rightarrow \mathbb{R} $. I think a proof by contradiction would work which means we want to prove $$\neg (\forall{y}\in ...
2
votes
2answers
92 views

Proving a BIG-O statement? Logarithmic expressions. Simple Induction.

I have to write a proof for the following statement. $$\log_2(n!)\in\mathcal O(n\log_2(n))$$ What approach would you recommend. I am kind of LOST trying to figure this out. I transformed the ...
1
vote
2answers
68 views

How can the following mathematical statements be proven?

I have these two mathematical statements: 1) $e^{i\pi}=-1$ and 2) $\ln(-1)=i\pi$. Now I want a proof of these statements. Can anyone help me proving these statements?
4
votes
2answers
79 views

Using Results in Sobolev Spaces

I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume $1 \leq p ...
1
vote
1answer
49 views

Proofs using vectors

I am entirely new to proofs, never done them for year 12, so I'm wondering how to solve these questions? This isn't homework, im preparing for an undergrad math olympiad on my own, so if you could ...
3
votes
2answers
40 views

Prove/refute: Every tautology is contingent

I'm asking to prove/refute the following statement: Every tautology is contingent. According to definition of contingent: A statement that is neither self-contradictory nor tautological is ...
1
vote
2answers
36 views

Reasoning why the implication $t - \epsilon \le x \le t + \epsilon$ for $\epsilon \ge 0 \Rightarrow x = t$ holds using sequences.

In texts I've seen the following reasoning used several times: Suppose $t - \epsilon \le x \le t + \epsilon$ holds for $\epsilon \ge 0$. Then it in particular holds for $t - \frac 1 n \le x \le t + ...
1
vote
1answer
44 views

Linear algebra: Matrix multiplication problem

I need to prove something in my homework I just don't know how to approach it and need some guidance. "Show that for a matrix $A$ ($n \times m$) and a vector $\vec{x}$ ($m \times 1$) it applies that: ...
1
vote
1answer
71 views

prove that if $L(f,P)=U(f,P)$ then $f$ is constant on $[a,b]$

Suppose that $f$ is a bounded function on $[a,b]$ and there exists a partition $P $of $[a,b] $such that $L(f,P)=U(f,P)$. Prove that $f$ is constant on $[a,b]$ I know that $L(f,P)=U(f,P)$ meaning $f$ ...
2
votes
6answers
66 views

Is the following True of False?

Provide a proof if true or a counterexample if false: Let a,b be two integers (not both zero), then the gcd(a,b) divides ay+bx for all for x,y ∈ Z. I tried with several cases such as gcd(5,10) = 5 ...
0
votes
3answers
46 views

Prove that if $F: A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is one-to-one

Prove that if $F: A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is one-to-one Proof: Suppose $F$ is not one-to-one. Then there exist $x_{1}, x_{2} \in A$ such that $F(x_{1}) = F(x_{2})$ where ...
0
votes
2answers
341 views

proving whether a function is one-to-one/onto

1) f(n) = 2n + 1 from set of integers to set of integers 2) f(n) = 2[n/2] from set of integers to set of integers [] is floor Could someone demonstrate how I ...
1
vote
1answer
75 views

Does $(\neg R\to R)\to R$ give rise to a proof strategy?

Take for example proof by contradiction, it can be viewed as a certain deduction in logic which can be used outside of logic to prove many interesting propositions. My question is: can we use $(\neg ...
1
vote
1answer
91 views

prove that $\int(f(x)+g(x))dx= \int f(x)dx+\int g(x)dx$

Let $f,g$ be two functions defined on $A$. Supposed that $F$ and $G$ are anti-derivative of $f$ and $ g$. Prove that $\int(f(x)+g(x))dx= \int f(x)dx + \int g(x)dx$ Here is what I got. Let $H(x)$ ...
2
votes
4answers
62 views

How do I prove a basic and obvious-looking set relations?

I'm a beginner in set theory, but the exercises asking for proof for intuitively obvious set relations like $A\cap A=A$. I don't know where to start. It will be appreciated if there is an example. ...
1
vote
0answers
63 views

Proving lower bounds from algorithmic game theory paper (specifically, price of anarchy is lower bounded by 3/2 for $m$ links)

This question is similar to Understanding proofs from paper on Game Theory (Price of Anarchy) This question is about the same proof: proving the lower bound that the price of anarchy (sometimes ...
1
vote
1answer
56 views

Prove that if $F : A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is Injective

Statement: if $F : A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is $1-1$ Proof: If $F$ is not $1-1$, then there exist $x_{1}, x_{2} \in A$ where $x_{1} \neq x_{2}$ and $F(x_{1}) = F(x_{2})$. ...
2
votes
0answers
40 views

Prove that $f: \mathbb{N} \rightarrow \mathbb{N}-\left \{ 1 \right \}$ given by $f(x) = x+1$ is $1$-$1$ and onto

$f: \mathbb{N} \rightarrow \mathbb{N}-\{1\}$ given by $f(x) = x+1$ is $1$-$1$ and onto. Proof: ($1$-$1$) Suppose $f(x_{1}) = f(x_{2})$ for $x_{1}, x_{2} \in \mathbb{N}$. Then $x_{1} + 1 = x_{2} + ...
0
votes
1answer
53 views

Proof by induction; simplify when adding k+1th term. Understanding induction.

I want to prove: $$(-\frac{1}{2})^0 + (-\frac{1}{2})^1 + \cdots + (-\frac{1}{2})^k + (-\frac{1}{2})^{k+1} = \frac{2^{k+1}+(-1^k)}{3\cdot2^k} + (-\frac{1}{2})^{k+1}$$ How do I simplify the last bit, ...
1
vote
2answers
43 views

Prove that if f is increasing on an interval I, then f is one to one on I.

How do I even begin to do this problem? I don't know where to even begin. The professor of the class tried to give us hints (as this is a redo to our homework) and said "The contrapositive is 'If f ...
0
votes
0answers
53 views

Help to understand manipulations on limits and integrals - $\int_a^b \! c \, \mathrm{d}x=c(b-a)$

I'm reading this proof from here: and I don't understand how to reach $$\lim_{n \to \infty} \left(\sum\limits_{i=1}^{n}c \right)\frac{b-a}{n}$$ Specifically, why are we allowed to take out ...
2
votes
4answers
70 views

Prove that for $n\ge 8$ there are nonnegative integers x and y s.t $3x+5y=n$

Prove that for every integer $n\ge 8$ there are nonnegative integers $x$ and $y$ such that $3x+5y=n$ Attempt: First of all I want to make it clear whether zero is a nonnegative integer. It ...
1
vote
1answer
58 views

Do this algorithm terminates?

Let $x \in \mathbb{R}^p$ denote a $p$ dimensional data point (a vector). I have two sets $A = \{x_1, .., x_n\}$ and $B = \{x_{n+1}, .., x_{n+m}\}$. So $|A| = n$, and $|B| = m$. Given $k \in ...
1
vote
1answer
132 views

Prove by minimum counterexample that $2^n>10n$ for $n>5$

Prove by minimum counterexample that for all integers $n>5$ the statement $2^n>10n$ is true. Attempt: Let $S$ be a set of counterexamples, $S=\{n \in \mathbb{Z_+}: 2^n \le 10n, \space ...
1
vote
3answers
58 views

When do two functions differ by a constant throughout an interval (Fundamental Theorem of Calculus)

I'm reading the proof of the Fundamental Theorem of Calculus here and I don't understand the following parts (at the bottom of page 2): I don't know how to conclude that $G(x)-F(x)=C$ for a $x \in ...
2
votes
1answer
47 views

What's the symbolic definition of the maximum value of a domain?

Lets say we have a domain S Maximum value of domain S = {S | ? ? ? ? ? ? } How could one define the possible maximum value of a set of values, symbolically?
1
vote
2answers
57 views

Prove that relation $R$ on a set of functions is an equivalence relation

Let set $S$ be the set of all functions $f:\mathbb{Z_+} \rightarrow \mathbb{Z_+}$. Define a realtion $R$ on $S$ by $(f,g)\in R$ iff there is a constant $M$ such that $\forall n (\frac{1}{M} < ...
1
vote
2answers
78 views

Prove that set of all points on a sphere is uncountable

Let $S=\{(x,y,z): x^2+y^2+z^2=4\}$ be the set of points on a sphere. Prove $S$ is uncountable. Attempt: Basically, each coordinate is between $0$ and $2$, i.e. $0\le x \le 2, 0\le y \le 2, 0\le z ...
2
votes
2answers
91 views

Prove that the x-axis in R^2 with the Euclidean metric is closed

I want to show that the x-axis is closed. Below is my attempt - I would appreciate any tips on to improve my proof or corrections: Let (X,d) be a metric space with the usual metric. WTS: {(x,y) | X ∈ ...
1
vote
3answers
34 views

Let $m$ and $n$ be integers in the ring of integers. Show that if $m\mathbb Z$ contains $n\mathbb Z$ if and only if $m$ divides $n$

Hello everyone working on the problem in the title...it's an if and only if proof so two directions to show start with the fact $n\mathbb Z$ is a subset of $m\mathbb Z$ and want to show $n=mx$ for ...
2
votes
1answer
289 views

Proving a constant function $f(x) = c$ is Riemann integrable

Prove that a constant function $f(x) = c$, where $c$ is in the Real Numbers, is Riemann integrable on any interval $[a, b]$ and $\int_a^bf(x) dx = c(b-a)$. By looking at the definition, it looks ...
0
votes
1answer
33 views

let $L_f, L_g, L_{f+g}$ be the lower integral of $f, g, and f+g$. Prove that $L_{f+g}\ge L_f+L_g$

let $f,g$ be two bounded function on $[a,b]$, let $L_f, L_g, L_{f+g}$ be the lower integral of $f, g, and f+g$. Prove that $L_{f+g}\ge L_f+L_g$ I don't know how to start
2
votes
3answers
431 views

Prove that if $f$ is integrable on $[a,b]$ then so is $|f|$?

Prove that if $f$ is integrable on $[a,b]$ then so is $|f|$. I can prove the converse of this is false, I also try using the definition of integrable function $f$, but I don't know what to do after ...
2
votes
1answer
77 views

Use induction on $n$ to prove that $2n+1<2^n$ for all integers $n≥3$.

Use induction on n to prove that $2n+1<2^n$ for all integers $n\geq 3$. My attempt: Let $P(n)$ be the statement $2n+1<2^n$. Base case: Prove that $P(3)$ is true. $LS = 2(3)+1=7$ and ...
0
votes
2answers
37 views
0
votes
3answers
186 views

Prove that set of all lines in the plane is uncountable.

Let $L$ be the set of all lines in the plane. Prove that $L$ is uncountable, but only countably many of the lines in $L$ contain more than one rational point. Attempt: Well, I was trying to ...
0
votes
2answers
89 views

Planar Graphs Question

I'm having some trouble with planar graphs. Two questions I was stuck on were: Prove that each planar graph on $n \gt 3$ vertices will have a minimum of $4$ vertices of degree $5$ at most. Let's say ...
2
votes
3answers
220 views

Proving if a function $f$ is differentiable and $f'(x)\ne0$ at all $x$, then it is one-to-one

Here's what the problem reads: Suppose that the function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and $f'(x)\neq 0$ for any $x \in (a,b)$. Prove that $f$ must be one-to-one. ...
1
vote
0answers
72 views

How to say this proof correctly: if d|a and d|b then d|a-b.

I believe I have this proof solved, but not sure that I wrote it correctly. Given that $d|a$ then there exist a $n$ such that $n = dk$ for some $k$ Given that $d|b$ then there exist a $m$ such that ...
3
votes
2answers
95 views

Argument over an induction proof

My friend gave me a problem. Define a sequence $<a_n>$ by the recurrence relation :$$ a_{n+2} - 6a_{n+1} + 8a_n = 0 $$ and $a_1 = 4, a_2 = 8 $. Find the general term $a_n$ in closed form. ...
2
votes
2answers
98 views

How to prove that if $-1<x<0$ then $x^2 + x < 0$?

I am trying to prove an equivalence. I have already proved that: $$x^2 + x < 0 \implies -1 < x < 0 $$ using a sub-proof by cases, in which I used the fact that when $xy < 0$, $x$ and ...
2
votes
1answer
68 views

Where is the error? Determining why a proof is incorrect.

Why is the following proof incorrect? I have tried to find out why for a few hours... The answer is: We aren't really proving the conclusion of each case is valid for the other case. Consider the ...
21
votes
10answers
1k views

Prove if $56x = 65y$ then $x + y$ is divisible by $11$

If $x$ and $y$ are natural numbers, and $56x = 65y$, prove that $x + y$ is divisible by $11$. I tried taking the $\gcd(56x,65y)$ using the Euclidean algorithm, but I got nowhere with it and do not ...
0
votes
1answer
61 views

Verify this equation (Proof help)

I'm trying to follow this book's equation, but I do not understand proofs. This is for my probability course book.
5
votes
5answers
249 views

Proving $n^3$ is even iff $n$ is even

I am trying to prove the following statement: Prove $n^3$ is even iff n is even. Translated into symbols we have: $n^3$ is even $\iff$ $n$ is even Since it's a double implication, I ...
-1
votes
5answers
542 views

Proving sets A and B are countable

a) Let A and B be disjoint sets, which are both countable. Prove that $A$ U $B$ is also countable. b) Use part (a) to show that the set of all irrational real numbers is not countable. So for part a ...
6
votes
2answers
107 views

Prove $(3x^2+3) \geq (x+1)^2+1$

$(3x^2+3) \geq (x+1)^2+1$ I tried using a direct proof but I think I got stumped along the way. $3x^2+3 \geq x^2+2x+2$ $2x^2+1 \geq 2x$ $2(x^2) +1 \geq 2x$ $x^2 + (1/2) \geq x$ How can I make ...