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
3answers
39 views

Why can't a direct proof be made backwards?

Say we have the following implication: $$\textit{Let $x\in \mathbb{Z}$. If $5x-7$ is even, then x is odd. }$$ The method used by my book to prove this implication is by means of a proof by ...
1
vote
0answers
15 views

Proposed proof for convergence in Sobolev space

Consider the Anisotropic Sobolev Space defined by: $$W^{1,\overrightarrow{p},\epsilon}(\Omega) := \{ u \in L^{1+\frac{1}{\epsilon}}(\Omega), \frac{\partial u}{\partial x_{i}} \in L^{p_{i}}(\Omega), ...
3
votes
2answers
24 views

Given a set $A$, how do I prove that there exists a set of all sets $x$ such that $\bigcup x=A$?

I am working with Zermelo-Fraenkel axioms. Specifically, I am allowed to assume the Axiom of Pair, Axiom Schema of Comprehension, Axiom of Union, and Axiom of Power Set, etc. (not yet allowed to use ...
1
vote
2answers
37 views

How to prove $x^{\phi(m)+1}\equiv x\pmod{p}$ [duplicate]

How do I prove that $x^{\phi(m)+1}\equiv x\pmod{p}$ when $m=pq$, two distinct primes? I kind of have an idea that it involves Euler's Theorem but it doesn't seem to be working as well as I wanted it ...
0
votes
1answer
41 views

Beginning Haskell - cannot understand proof

I've just started reading "Thinking Functionally with Haskell" by Richard Bird In the preface he states : And after stating the proof he also states the proof will be used throughout the book. ...
-5
votes
1answer
34 views

Consider the expression n^5 + 9n. [on hold]

a) Prove directly that $n^5 + 9n$ is even for all $n \in \Bbb N$ b) Prove by induction that $n^5 + 9n$ is divisible by $5$ for all $n \in \Bbb N$ c) Prove that for all $m \in \Bbb N$, $2 \mid m$ and ...
-4
votes
0answers
29 views

Let $f: ℝ \to ℝ$ be the function given by $f(x)=2x-3$. Then $f$ is onto. [on hold]

I need help proving or disproving this statement. Let $f: ℝ \to ℝ$ be the function given by $f(x)=2x-3$. Then $f$ is onto.
1
vote
1answer
16 views

show that a loopless graph $G$ contains a bipartite spanning subgraph $H$ such that $d_H(v) \ge \frac{1}{2} d_G(v)$ for all v $\in$ V.

The hint in the appendix of book says that bipartite subgraph with with largest possible number of edges has such a property, but I don't know how to use this hint! any help would be appreciated.
0
votes
1answer
34 views

Calculus proof attempt

Why does $$\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{\dot{y}}{\dot{x}}\right)\frac{1}{\dot{x}}$$ not yield the same result as $$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{\dot{x}}\right) \ ...
0
votes
1answer
23 views

second derivative of a parametric equation

can someone please explain how in the proof for the second differential of a parametric function we get from to ? how do we calculate $\frac {d}{dt}$?
0
votes
2answers
29 views

Induction proof for $x \le y $

So these are $x \in \{1 ... i\}$ and $y \in \{ i + 1 ... n\}$, for $i, n \in \mathbb N$. I want to prove it for every $x \le y $. I know it`s easy but the solution is escaping me. I have tried with ...
-2
votes
2answers
45 views

Please help prove this summation problem for me [on hold]

Prove that for all integers n greater than or equal to 1, $\sum_{k=1}^{3n} (4k+3)=3n(6n+5)$.
1
vote
1answer
34 views

Prove or disprove this lemma for Catalan Numbers

Prove or disprove that for all non-negative integers $n$ and $r$ with $r+1$ is less than or equal to $n$, $C(n,r+1)=C(n,r)\times\frac{n-r}{r+1}$.
0
votes
0answers
30 views

How to prove that all powers of two minus one have only 1's when in binary representation?

It just came to my mind that all powers of two, when represented in binary format, are composed of only 1's, not 0's. I can see some logic behind it, however I can't seem to come up with an actual ...
2
votes
2answers
43 views

$X:\Omega \to \mathbb{N}$ is random variable, How to prove that $E[x]=\sum_{i} \Pr(X\ge i)$?

I'm stuck with this proof: $X:\Omega \to \mathbb{N}$ is random variable, prove that $\mathbb{E}[X]=\sum_{i=1,2,3...} \mathbb{P}(X\ge i)$? How I'm proving it? I'm starting with the definition: ...
1
vote
2answers
38 views

How does this proof that the square of an integer, not divisible by 5, leaves a remainder of 1 or 4 when divided by 5 work?

Below I have a part of a proof of the fact that the square of an integer, not divisible by $5$, leaves a remainder of $1$ or $4$ when divided by $5$. But I am wondering where does the part highlighted ...
6
votes
2answers
51 views

How do I close the gap between intuitively knowing something is true vs being able to prove it?

For example, one of my review problems is: Let $S_k$ be the kernel of $T^k$. Show there is a $K$ such that $S_K = S_{K+1} = \cdots$ Somewhere in the back of my brain there's an intuition that told ...
1
vote
1answer
25 views

Proving if $\frac{3x+1}{x-1}$ is onto?

So, I have this function: $f(x)=\frac{3x+1}{x-1}$. So, in proving if it is onto, then by definition, for every b in B, there exists an a in A such that $f(a)=b$. So, let's solve or a. We get: ...
1
vote
1answer
29 views

How many square root matrices?

I would like to see a proof of the following statement: A positive-semidefinite matrix has precisely one positive-semidefinite square root, which can be called its principal square root. I ...
1
vote
1answer
43 views

Consider the function $g(x)=xe^x$. Make and prove a conjecture about the $n^\text{th}$ derivative of $g$. [closed]

Please help on this homework problem I have in my Proofs class. My prof is really bad at explaining and I don't know how to answer this problem! Thank you!
-5
votes
0answers
14 views

Suppose that R and S are both relations on set A. Prove or disprove any two of the following: [closed]

a) R ∩ S is transitive if either R is transitive or S is transitive. b) R ∪ S is symmetric if R and S are both symmetric. c) If R\S is reflexive, then R is reflexive or S is reflexive. How to I ...
0
votes
3answers
51 views

Prove that a rational number minus an irrational number must be irrational. [duplicate]

Please help with this homework problem I have! I don't know how to prove this.
0
votes
1answer
21 views

Can you use proof by contradiction inside simple induction?

During a proof using simple induction, I assumed P(k) is true. Now in order to show P(k+1) is true using P(k), can I do a proof by contradiction on P(k+1) and say P(k) would be wrong if P(k+1) is ...
-1
votes
1answer
30 views

Can I prove these series with limit a using induction?

This is the equation: It is true for: E a normed space and $(a_n)_{n \in \mathbb N}$ a convergent sequence with limes a. $$s_k = \frac1k\sum^k_{n=1} a_n \rightarrow a$$ $a = \lim_{n\rightarrow ...
1
vote
0answers
40 views

Prove formula's tautology

Prove that a formula that only consists of variables, logical negation and logical equality, and in which all variables and negation appear for an even number of times, must be tautological.
0
votes
2answers
29 views

Formal negation of $((p\rightarrow q) \vee (q \leftrightarrow r)) \rightarrow q$

Can someone give me an outline for how I can negate the following? $((p\rightarrow q) \vee (q \leftrightarrow r)) \rightarrow q$
3
votes
2answers
68 views

Integer inequality: $x + y +z> a + b + c$ does not imply $xyz > abc$

Prove by contradiction that for any integers $x,y,z,a,b,c$ greater than $0$ such that $x+y>a+b$, it is not implied that $x\cdot y\cdot z>a\cdot b\cdot c$? Obviously this statement is true. ...
2
votes
2answers
46 views

Alternate way to Prove or disprove $6\mid n(n+1)(n+2)$

This is my proof, I'm wondering if I'm correct, and how to do without induction. My Work Basis Step $$\frac{(1)(2)(3)}{6} = 1$$ Inductive Hypothesis Assume that $\dfrac{k(k+1)(k+2)}{6} = d$ where ...
1
vote
3answers
39 views

Explain proof that any positive definite matrix is invertible

If an $n \times n$ symmetric A is positive definite, then all of its eigenvalues are positive, so $0$ is not an eigenvalue of $A$. Therefore, the system of equations $A\mathbf{x}=\mathbf{0}$ has no ...
1
vote
1answer
34 views

Let A⊆R be a set. Prove that A is bounded if and only if there is some M∈R such that M>0 and that |x|≤M for all x∈A.

I proved the above problem as follows but received feedback that I'm not certain I understand. Can someone help me determine where I went wrong in the proof? Let A⊆R be a set. Suppose M∈R where ...
0
votes
2answers
47 views

Proof for consecutive integers

Prove that if $n$ is an odd integer, $n^3$ is the sum of $n$ consecutive integers. I'm confused on how to prove something with consecutive integers.
0
votes
1answer
28 views

Greatest Common Divisor Proof

Show that if $r_k = q_i r_{k+1} + r_{k+2}$, then $\gcd(r_k,r_{k+1}) = (r_{k+1},r_{k+2})$
0
votes
4answers
41 views

Induction Proof with a $\neq$ 1 [duplicate]

For $a \neq 1$ and $n>0$, $(1-a^{n+1})/(1-a)=1+a+a^2+...+a^n$ How do you prove this by induction?
2
votes
1answer
20 views

Prove an x exists with f(x) = f(x + T/2)

Suppose $f: \mathbb{R} → \mathbb{R}$ is a continuous and periodic function with period $T > 0$. Show that: there exists an $x \in \mathbb{R}$ such that $f(x) = f(x + T/2)$. We figured out we ...
1
vote
3answers
71 views

Derivative=0 implies double root

Let a be the root of a polynomial f(x) and let f'(a)=0. Then a is a double root of f(x). What is the name of this theorem and does someone know a simple (high school level) proof?
0
votes
3answers
44 views

Prove/disprove: If $n\in \mathbb N$ with $n>2$ not prime, then $2n + 13$ is not prime.

Prove/disprove: If $n\in \mathbb N$ with $n>2$ not prime, then $2n + 13$ is not prime. From the context in which this question was set, I believe I have to prove/disprove it using ...
0
votes
1answer
38 views

Solve by induction.

How would I show this equation is odd by using the induction hypothesis: $$ g(s) = 3(g(s-1))+(g(s-2))+1 $$ I was thinking that I would prove $g(s)$ is odd by $g(s+1) = 3(g(s)+g(s-1))+1$. How would I ...
0
votes
0answers
29 views

trouble in reading statements involving symbols

I have trouble in reading statements involving symbols. (1) For example, when reading a statement from a paragraph: Suppose the size of the set is n. The symbol ...
1
vote
1answer
18 views

Proof by induction for a recursive function f

Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive case: $\operatorname{f}(x) = ...
0
votes
2answers
36 views

Prove that $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$

I'm trying to practice proof writing, and found the following text question: For all sets A,B,C: $ (A \cup B) \cap C \subseteq A \cup (B \cap C)$ The first step I was thinking of showing is that: ...
2
votes
0answers
42 views

Assuming a Solution Exists

Students who are beginning to learn proofs (and some seasoned pros) occasionally commit the error of assuming what they're trying to prove. My question involves assuming at the onset that a solution ...
1
vote
1answer
22 views

If (an)→ L, an > 0 for all n ∈ N, and L > 0, then prove that √an → √L .

To be honest, I don't even understand what the question is asking, and have no idea how to answer it. Any guidance would be great. I know convergent/divergent definitions, as well as basic limit laws, ...
0
votes
3answers
30 views

Disproving statement of convergence

How can I disprove this statement: If the sequence $(a_n+b_n)^\infty_{n=1}$ converges, then both $(a_n)^\infty_{n=1}$ and $(b_n)^\infty_{n=1}$ converge. Is this statement able to be disproved? I ...
0
votes
4answers
35 views

Suppose that the function $f: \mathbb {R}\to\mathbb{R}$ is continuous. Define $G(x) = \int_{0}^x (x - t)f(t) dt$, $\forall x$ [closed]

Suppose that the function $f: \mathbb {R}\to\mathbb{R}$ is continuous. Define $G(x) = \int_{0}^x (x - t)f(t) dt$, $\forall x$. Prove $G''(x) = f(x)$ for all $x$.
2
votes
1answer
29 views

Trouble Understanding Proof About Polynomials

In the question I have to prove that: There is no polynomial $P (x) = a_n x^n + a_{n−1}x^{n−1} + · · · + a_0$ with integer coefficients and of degree at least 1 with the property that $P(0), ...
1
vote
3answers
47 views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
0
votes
1answer
38 views

Prove the sequence $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge

How would I be able to prove that $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge? I know that for a sequence to converge to a limit, then for all $\varepsilon > 0, \exists N \in \mathrm N ...
0
votes
1answer
26 views

Proof by induction for a recursive function

Really having a tough time doing this question: Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive ...
1
vote
1answer
55 views

Solve this tautology

Hypotheses: not $q$, $p$ or not $s$, $p \rightarrow$ ($d$ and $q$), $e \rightarrow s$ Conclusion: not $e$ I have thus far, but unsure how to proceed. I am looking forward to solve it using ...
0
votes
2answers
62 views

Prove that $x=0.1234567891011\cdots$ is irrational [duplicate]

Prove that $x=0.1234567891011\cdots$ is irrational Proof: we argue by contradiction.suppose x is rational. then its decimal expansion ultimatetly periodic. Lets p denote the perid of this expansion. ...