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.
0
votes
2answers
46 views
Let $R$ be an equivalence relation on a set $A$, $a,b \in A$. Prove $[a] = [b]$ iff $aRb$.
Hello I need help with the proof strategy for this problem.
Let $R$ be an equivalence relation on a set $A$ and let $a,b \in A$. Prove that $[a] = [b]$ if and only if $aRb$.
0
votes
0answers
21 views
Hamiltonian cycle problem: how to prove NP-completeness?
How to prove that finding a Hamiltonian cycle in a graph is an NP-complete problem? Should I try to reduce the travelling salesman problem (TSP) to this one (Hamiltonian cycle)?
4
votes
2answers
32 views
Prove that one of the following sets is a subspace and the other isn't?
OK, here goes another.
Prove that $ W_1 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + \cdots + a_n = 0$} is a subspace of $F^n$ but $ W_2 = ${$(a_1, a_2, \ldots, a_n) \in F^n : a_1 + a_2 + ...
0
votes
1answer
44 views
Is this proof complete? Case of a metric space
I am looking to prove whether or not the following statement is true:
Let $M$ be a set and $D_1$ and $D_2$ metrics on M such that they form a metric space. Then $(M, \min(D_1 , D_2))$ forms a ...
0
votes
3answers
63 views
Proof regarding factorials.
Suppose $a$ and $k$ are positive integers, then how would you prove(not intuitively) that:
$a!k! \leq (ak)!$
Although it is apparent that the inequality is correct, but how can I show this ...
0
votes
4answers
62 views
$a^{\log_b(c)} = c^{\log_b(a)}$ [duplicate]
I'm not sure how to start. My questions is how do you prove:
$$a^{\log_b(c)} = c^{\log_b(a)}$$ where $a,b,c > 0$.
1
vote
3answers
43 views
$3^{3n+1} < 2^{5n+6} $ for all non-negative integers $n$. Is my induction solution correct?
Show using mathematical induction that $3^{3n+1} < 2^{5n+6} $ for all non-negative integers $n$. I'm not sure whether what I did at the last is valid?
Basis step:
for all non-negative integers
...
1
vote
3answers
23 views
What do you use for your basis step when its domain is all integers?
Example: *For all integers$
n
, 4(
n
^2
+
n
+
1)
–
3
n
^2$
is a perfect square what should i use? negative infinity?
I know you can use a direct proof but what if theres an induction question with ...
1
vote
1answer
75 views
$G$ is a tree $\iff G$ contains no cycles, but if you add one edge you create exactly one cycle - is my proof correct?
$G$ is a tree $\iff G$ contains no cycles, but if you add one edge you create exactly one cycle
Could anyone please be so kind to check my proof? That would be very much appreciated. Thank you in ...
2
votes
0answers
36 views
Checking an inductive proof on a combinatorial product
Consider the following product, for $n, k, i \in \Bbb Z_+, k \geq 2$:
$$
{\prod_{\ell = 1}^i {n + k - \ell \choose k } \over \prod_{\ell = 1}^{i-1} { k + \ell \choose k}} \tag{$*$}
$$
It has been my ...
0
votes
1answer
122 views
Mathematican proof or physicist proof of a theorem?
I am writing a document, where I have proved a so-called theorem coming from a physicist paper. It is not a mathematical theorem in the sense that have only done the computations and did not check ...
2
votes
0answers
79 views
Proving there are no integer solutions for $3x^2=9+y^3$
Prove there are no $x,y\in\mathbb{Z}$ such that $3x^2=9+y^3$.
Initial proof
Let us assume there are $x,y\in\mathbb{Z}$ that satisfy the equation, which can be rewritten as $$3(x^2-3)=y^3.$$ So, ...
3
votes
1answer
41 views
Equivalence relation - Proof question
Prove that the relation, two finite sets are equivalent if there is a one-to-one correspondence between them, is an equivalence relation on the collection $S$ of all finite sets.
I'm sure I know the ...
1
vote
4answers
83 views
Standard logic notation in mathematics
My profesor is always complaining that my proofs are very long and difficult to read because I never use notation, meaning I say everything in words. Tired of that I decided to study logic by myself ...
1
vote
0answers
71 views
What are universally accepted qualities that constitute great mathematical exposition? [closed]
It seems almost everyone is in agreement that Serre and Halmos are great expositors of mathematics. Looking at a monograph written by Serre ("A course in Arithmetic"), I noticed his proofs were very ...
2
votes
3answers
45 views
Need help with Cantor-Bernstein-Schroeder Proof at ProofWiki
This concerns Proof 6 of the CBST theorem at ProofWiki.
I am stuck on the line beginning "Similarly, let $g' = $"
The 2nd equality on this line is not immediately obvious to me. How do you prove ...
-1
votes
0answers
28 views
Proof in Probability Distribution Functions
The number of heads a coin comes up tails when tossed n times is denoted by random
variable X. Suppose that for each toss, the coin will appear heads with probability z.
(a) The probability mass ...
0
votes
3answers
63 views
Prove that $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ converges when $n \to \infty$
I want to prove that the sequence defined by $\{x_1=1,\,x_{n+1}=\frac {x_n}2+\frac 1{x_n} \}$ has a limit.
By evaluating the sequence I notice that the sequence is strictly monotonically decreasing ...
9
votes
3answers
110 views
Please review my question and solution. Thanks in advance.
How many values of x are there such that there exists positive integer solutions for S, such that $S=\sqrt{x(x+p)}$ where $x$ is an integer and $p$ is a prime number $>2$
This is a problem I made ...
-4
votes
0answers
47 views
Requesting clarification with my problem. [closed]
How many values of x are there such that there exists positive integer solutions for S, such that $S=\sqrt{x(x+p)}$ where $x$ is an integer and $p$ is a prime number $2$
This is problem I made and ...
3
votes
6answers
67 views
Is $\sum_{i=1}^{n-1}i=\binom{n}{2}$?
How can I show that
$$
\sum_{i=1}^{n-1}i=\binom{n}{2}?
$$
This is what I have tried, but I do not know if it is correct:
Proof.
Let $n=2$. Then,
$$
\begin{align}
\sum_{i=1}^{1}i&=1\text{, ...
0
votes
1answer
47 views
Could someone help me to improve the proof writing?
I will prove the following claim. I'm not a native English speaker. Could someone help me to improve the writing?
A regular pseudocompact Moore space is ccc and first countable.
Prove: I will ...
1
vote
1answer
186 views
How much room is there for original mathematics research?
Should more universities and colleges offer degrees in mathematical research?
I am in the process of incorporating a non-profit college and I am considering offering a degree in mathematical ...
2
votes
5answers
81 views
For $a, b \in \mathbb Z,\;$ if $\;a^2(b^2-2b)$ is odd, then a and b are odd. Proof check.
Suppose $a,b$ are integers, if $a^2(b^2-2b)$ is odd, then a and b are odd, is my solution the best way?
PS: I know this is easy but do i need to expand the final answer? because im practicing for ...
0
votes
1answer
20 views
Definition by Recursion and a Question about Induction
I have some questions to ask.
Suppose I want to define some sequence of propositional formulas $\{\varphi_{j}\}_{j\in\mathbb{N}}$. First, I define it this way. Fix an enumeration ...
2
votes
1answer
78 views
A question about quantifiers
I'm trying to prove this theorem:
Let $F$ and $G$ be functions. Then $F=G$ if and only if $\operatorname{Dom}(F)=\operatorname{Dom}(G)$ and $\forall X (X\in \operatorname{Dom}(F)\rightarrow ...
0
votes
0answers
22 views
Prove limit using $\epsilon-M$ prove, To show $\lim_{n->\infty} \frac{n^2+2n}{n^3-5}$
To show $\lim_{n->\infty} \frac{n^2+2n}{n^3-5}$so here is my approach, Im not sure about something, just work out the preproof here, So I will first guess limit is 0, then start my preproof
...
2
votes
1answer
22 views
Linear Algebra dependent Eigenvectors Proof
Problem statement:
Let $n \ge 2 $ be an integer. Suppose that A is an $n \times n$ matrix and that $\lambda_1$, $\lambda_2$ are eigenvalues of A with corresponding eigenvectors $v_1$, $v_2$ ...
5
votes
1answer
117 views
British maths style guide
For British maths style, is this punctuation OK?
so if $x=-3$, then $\left|x\right|=3$, and if $x=7$, then $\left|x\right|=7$, etc
with commas before "then" and "and".
1
vote
2answers
50 views
Proving a biconditional statement with an or
I want to prove a theorem in geometry of the form $p \iff q \vee r$. My plan is to prove:
$q \implies p$ as well as $r \implies p$
$p \text{ and } \lnot q \implies r$
Can I get someone to verify ...
0
votes
3answers
40 views
$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x\log \pi + (n-x)\log(1-\pi)\;\;?$
$$\log\left(\binom{n}{x} \pi^x (1-\pi)^{n-x}\right)=x \log \pi + (n-x)\log(1-\pi)$$ this is what i have.
i dont understand how $\binom{n}{x}$ disappears, but the rest is fine.
I tried this, but it ...
4
votes
1answer
90 views
How can I prove $2\sup(S) = \sup (2S)$?
Let $S$ be a nonempty bounded subset of $\mathbb{R}$ and $T = \{2s : s \in S \}$. Show $\sup T = 2\sup S$
Proof
Consider $2s = s + s \leq \sup S + \sup S = 2\sup S $. $T \subset S$ where T is ...
2
votes
1answer
87 views
What should be proved in the binomial theorem?
I'm following Cambrige mathematics syllabus, from the list of contents of what should be learned:
Induction as a method of proof, including a proof of the binomial theorem with non-negative ...
3
votes
2answers
69 views
Image of closed unit ball under a compact operator
Let $X,Y$ be Banach spaces and $A\in\mathcal L(X,Y)$ . The task is to prove the following:
$A$ is compact if and only if the image of the closed unit ball in $X$ is compact in $Y$.
I have proven ...
3
votes
4answers
88 views
Prove $a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1$ without use of log properties
$a^n \rightarrow 0$ as $n \rightarrow \infty$ for $\left|a\right| < 1 $
Hint $u_{2n}$ = $u_{n}^2$
I have totally no idea how to prove this, this looks obvious but i found out proof is really ...
1
vote
1answer
40 views
Continuity and limits. Please check epsilon delta
Suppose $f$ is continuous at $a$ and $f(a) = 0$. Prove that if $\alpha \neq 0 $, then $f+\alpha$ is nonzero in some open interval containing $a$.
Since $f$ is continuous, we take $\epsilon = ...
3
votes
2answers
67 views
What are some examples of proof by contrapositive?
Applying the Modus Tollens argument to Fermat's Little Theorem really helped me to understand logical implication. I never knew that FLT was actually a compositality test.
Theorem (FLT): given ...
5
votes
2answers
82 views
natural language proof assistant
I was wondering whether there has been any attempt to create a proof assistant that you write in it, in english,
I mean you write your proof the usual way in TeX(maybe use a 'simpler english') then ...
1
vote
3answers
61 views
How do I prove that the sine function on the domain $[-1/2, 1/2]$ is injective?
How do I prove that the sine function on the domain $[-1/2, 1/2]$ is injective?
I completely understand the concept but I'm having trouble writing a proof for this. Thanks in advance.
8
votes
3answers
157 views
Intuition of Addition Formula for Sine and Cosine
The proof of two angles for sine function is derived using $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ and $$\cos(A+B)=\cos A\cos B-\sin A\sin B$$ for cosine function. I know how to derive both of the ...
9
votes
3answers
210 views
Prove by Hilbert deduction: $\vdash _{HFOL} \forall x (\neg(A \to \bar{B}))\to \neg(\forall xA \to \neg(\forall xB))$
I'd really like your help proving:
$\vdash_{HFOL} \forall x (\neg(A \to \bar{B}))\to \neg(\forall xA \to \neg(\forall xB))$
Where $HFOL$ is the proof system which contains the Hilbert relevant ...
2
votes
3answers
77 views
Show that a vector that is orthogonal to every other vector is the zero vector
I have the following question, and I'd like to get some tips on how to write the proof. I know why it is, but I'm still not so great at writing it mathematically.
If $u$ is a vector in ...
0
votes
1answer
35 views
How do I prove $\dim{U} = \dim{W}$ when…
$\mathbf{U} = sp\{(a_1 a_2 a_3),(b_1 b_2 b_3),(c_1 c_2 c_3)\}$
$\mathbf{W} = sp\{(a_1 b_1 c_1),(a_2 b_2 c_2),(a_3 b_3 c_3)\}$
$\mathbf{U}$ and $\mathbf{W}$ are subspaces of $\mathbb{R}^3$
0
votes
0answers
50 views
property of an increasing or decreasing function
For $x \in \mathbb{R}$, and $f(x)$ an increasing function, can we prove whether
$$ af(x)\lesseqgtr f(ax) $$
for $a >0$? If we have additional information that $f$ is homogeneous of some degree, ...
3
votes
0answers
184 views
Proving identity involving sum
I'm stuck trying to prove the following identity:
$$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose j}(4S-i-k-3)!(4S-2i-1)}{(4S-i-j-1)!} $$ $$=\frac{(-1)^{j+k}(4S-2k-3)!{k+1 ...
0
votes
2answers
156 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
39
votes
14answers
2k views
What is a proof?
I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra).
Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
2
votes
3answers
59 views
problem with induction?
I am a bit new to logical induction, so I apologize if this question is a bit basic.
I tried proving this by induction:
$$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$
Starting with the base ...
1
vote
2answers
77 views
Law of large numbers?
Given random variables $Z_1,Z_2,Z_3,\ldots$, which are uniformly distributed for $[8,10]$:
If $X_k =\min\{Z_1, Z_2,Z_3,\ldots,Z_k\}$, prove convergence in probability and find the constant. ...
0
votes
2answers
51 views
If $E = \{ x \in \mathbb{R}: \sin(\frac1{x}) = 1\}$ then $l = 0$ is a limit point of E
If $E = \{ x \in \mathbb{R}: \sin\left(\frac{1}{x}\right) = 1\}$, then $l = 0$ is a limit point of $E$.
I have a proof here but I don't quite understand a few points, I hope someone can explain it a ...
