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

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3
votes
3answers
68 views

Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? [closed]

As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and ...
7
votes
3answers
444 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
0
votes
2answers
51 views

On proving $(f^{-1})'(b) = \frac{1}{f'(a)}. $ where $b = f(a)$.

Could somebody kindly provide a proof or a reference to a proof of this fact: Let $ I $ be an open interval, and suppose that $ f: I \to \mathbb{R} $ is one-to-one and continuous on $ I $. If $ f ...
1
vote
2answers
52 views

Proof in set theory

Let $A,B,C$ -- subsets in some fixed set. Prove that $A \cap B \subseteq C$ iff $A \subseteq \overline{B} \cup C$. Have no ideas how to prove this. On the language of definitions we have $$x ...
4
votes
2answers
213 views

Proof of determinant formula

I have just started to learn how to construct proofs. That is, I am not really good at it (yet). In this thread I will work through a problem from my Linear Algebra textbook. First i will give you my ...
1
vote
2answers
78 views

Proving that $\hat{x} = x$ in least square

I am trying to show that if $Ax=b$ has a unique solution $x$, then the least square solution $\hat{x}$ is the exact one (i.e., $\hat{x} = x$). My attempt: We know that for $A x = b$ to have an exact ...
1
vote
3answers
189 views

Prove that there are infinity many tautologies.

For this question I think I am suppose to use proof by contradiction, but I need some hints on how to proceed with the proof. Always if someone can give me a brief explanation on how proof by ...
2
votes
4answers
271 views

How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
1
vote
2answers
87 views

Prove that $\sin^{2}{\theta} + \cos^{2}{\theta} = 1.$

I believe that I have been able to prove that Prove $\sin^{2}{\theta} + \cos^{2}{\theta} = 1, \forall \theta,$ but I would like to ask if my proof is correct / valid.
2
votes
1answer
33 views

Show that $dim(X,\succeq)\leq |X^2|$ when $X$ is finite

I am trying to prove that when $(X,\succeq)$ is a finite preorder, the $dim(X,\succeq)\leq |X^2|$. Here's the full context (Exercise 11 (a)): My idea of resolution was to show that any set of ...
1
vote
2answers
53 views

Path and cycle length proof

How can we prove this proposition : Every graph G contains a path of length $ \delta(G)$ and a cycle of length at least $ \delta(G) + 1$, provided that $\delta(G) \geq 2 $. ($ \delta(G)$ is the ...
1
vote
1answer
72 views

prove that $p(n) := n^2 + n + c$ is not prime

The question is in MIT Mathematics for CS assignments but unfortunately there is no solutions. -> I do understand that it is false if we use $n = c$ or $n = c-1$ but cannot formally write it as ...
4
votes
5answers
112 views

Prove every integer is of the form $5k+r$ with $0\le r<5$

I have came across this question from my text book: Prove or disprove: any integer $n$ is of the form: $5k$, $5k + 1$, $5k + 2$, $5k + 3$ or $5k + 4$ for some integer $k$. I'm not sure what would be ...
0
votes
0answers
69 views

Comparison of books that teach proof techniques

I have to take discrete math and want to learn proof techniques both to get ahead in it as well as open up the possibility of understanding higher math. I've seen several books recommended such as How ...
1
vote
1answer
53 views

Primary decomposition of modules - uniqueness proof

Let $M$ be $A$-module, $A$ commutative ring, and $N$ submodule and let $$N=Q_1\cap\dots\cap Q_r=Q'_1\cap \dots \cap Q'_s$$ be reduced primary decompositions of $N$. Then $r=s$. The set of primes ...
2
votes
3answers
71 views

Proof for $\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$

Problem: For positive integers $n,j,k$, prove that the following holds: $$\left\lfloor\frac 1j\left\lfloor\frac nk\right\rfloor\right\rfloor=\left\lfloor\frac n{jk}\right\rfloor$$ I simply ...
1
vote
3answers
62 views

How to prove that the cross product of a countable and uncountable set is uncountable?

so my question is, how can you prove that ${\Bbb Z}$ x ${\Bbb R}$ is uncountable? So far I have tried proving that there is an uncountable subset of ${\Bbb Z}$ x ${\Bbb R}$ without luck and I'm ...
1
vote
1answer
100 views

derivative $1 \over x$ -proof

proving $\frac{1}{x}$ by definition $$\left(\frac{1}{x}\right)'=\lim_{h \to 0} {\frac{1}{x+h}-\frac{1}{x}\over h}=\lim _{h \to 0} {\frac{x-x-h}{(x+h)x}\over h}=\lim _{h \to 0} ...
0
votes
2answers
40 views

How to prove this product rule?

If $f,g:\Omega\subset\mathbb{R}^n\rightarrow\mathbb{R}$ are differentiable in $x_0\in\Omega$ ($\Omega$ is open), then the function $(f*g)$ is differentiable in $x_0$ and: $(f\cdot ...
1
vote
1answer
158 views

proof of rational numbers as repeating or terminating decimald

As an exercise in my conceptual algebra class we attempted to determine the reason why this theorem holds true in the forward direction. (Note we decided not to tackle the opposite direction) I wrote ...
1
vote
1answer
63 views

Methods to prove that a function is continuous

Although I seem to understand the concept of continuity in connection with functions, I am often stuck proving that particular functions are continuous. I think the epsilon-delta definition is the ...
1
vote
1answer
26 views

Showing there exists a complex differentiable function $g$ satisfying $g(z_0)=z_0$, with $g'(z_0) \neq 0$ and that $h(g(z))=(z−z_0)^{−m}$.

This is a follow up to a previous question: (Supposing $h$ has a pole, order m, at $z_0$, show the existence of a neighbourhood of $z_0$ and a new complex differentiable function $g$.) I'm trying to ...
0
votes
1answer
139 views

Prove that if function f is monotonic, then it one-to-one

What I have so far: Suppose $f$ is monotonic. It is therefore either increasing or decreasing. Proof for increasing: If $f$ is increasing, then $f(x_1) <f(x_2)$ whenever $x_1 < x_2$, which ...
1
vote
0answers
46 views

Help in Understanding the Proof of Baire-Category theorem

In the proof of the Baire category theorem(for non-empty Banach Spaces), I cannot understand the following Baire Category Theorem: A non-empty Banach Space cannot be a countable union of nowhere ...
2
votes
3answers
109 views

Induction Proof: $\sum_{k=1}^n k^2$

Prove by induction, the following: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}6$$ So this is what I have so far: We will prove the base case for $n=1$: $$\sum_{k=1}^1 1^2 = \frac{1(1+1)(2(1)+1)}6$$ We ...
0
votes
0answers
23 views

Reasoning in designator/formula proof

(Boldface letters denote syntactical variables.) Claim: Consider the formula $\lor \mathbf w\mathbf r$ in a first order language, where $\mathbf w$ and $\mathbf r$ are formulas. We have that $\lor ...
0
votes
1answer
41 views

Euclid's elements proposition 13 book 3

"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within ...
0
votes
0answers
30 views

A way to prove the focal property of the ellipse

Every light ray which is radiant from a focal point reflects on the ellipse, such that it goes through the other focal point. Assuming $P=(x_0,y_0)$ is an arbitrary point of an ellipse with the ...
3
votes
2answers
37 views

proof with complex integration by u-substitution

If $f$ is continuous in $[0,\pi]$, use the substitution $u = \pi - x$ to show that $\int_0^{\pi} xf(\sin x)dx = \frac{\pi}{2}\int_0^{\pi} f(\sin x)dx$ Not having much idea where to begin, I ...
-5
votes
1answer
158 views

I have proved that 1 + 1 = 0 [closed]

I have proved that 1 + 1 = 0 in one of my questions where a field was given. I was wondering if it is true in every field we have 1 + 1 = 0. Also i was wondering (i know how to prove 1 + 1 = 0) can ...
-1
votes
2answers
96 views

Every field has at least two elements

I got a question saying in every field (F, +, ⋅, 0, 1), the set F has at least 2 elements. It asks if it is true prove it or if false provide a counterexample. I understand the idea of finite fields ...
1
vote
2answers
56 views

Field Question Proof with Axiom 4

Prove that if $(F,+,⋅,0,1)$ is a field, then there is no element $w ∈ F$ such that $0 \cdot w = 1$. Note that Axiom 4 from lecture (aka "M4" in the textbook) ensures that for $x ≠ 0$, there is a $w ∈ ...
1
vote
1answer
37 views

Can I use logical equivalence instead of biconditional in proofs?

My textbook defines the symbol <=> to mean equivalent to, has the same solutions as or if and only if. It defines the symbols => and <= to mean implies or leads to. The textbook does not use the ...
1
vote
1answer
41 views

Field Question Proofs

True or False: In every field $F$, if $x,y$ belong to $F$ and $w,w'$ belong to $F$ such that $x * w = 1$ and $y * w' = 1$, then $(x * y) * (w * w') = 1$. I think the answer would be false mainly ...
3
votes
2answers
61 views

Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $.

Define $S\equiv\{ x\in \mathbb{Q}\mid x^2<2\}$. Show that $\sup S=\sqrt{2} $. For this question, I think that I would use the completeness axiom. As $3$ is greater than $2$, so $S$ has a ...
-1
votes
1answer
113 views

f and g are bounded with domain of integers and target the real numbers . If f/g is bounded, then g/f is bounded.

I have come up with two bounded functions f = 1/x^2+1 and g = 1/x^2+2 and these tell me that g/f is also bounded. However, I am having trouble writing a proof or proving that g/f is not bounded by ...
-1
votes
1answer
150 views

f and g are bounded . if 1/g is bounded, then f/g is bounded.

I would like some help understanding how to go about this question. I think that f/g is not bounded, but I cannot figure how to show that f/g is not bounded.
3
votes
2answers
148 views

Prove $\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$ using $\delta-\epsilon$ definition of limit

I need to prove that the $$\lim_{x\to3}\frac{x^2 - 9}{x - 3} = 6$$ using $\delta-\epsilon$ definition of limit. Now, I have started with a discussion, saying that what we want is that if $\left| x - ...
13
votes
4answers
740 views

How to learn/speak “mathematical english”?

Good day! I was wondering if there is a good way to learn "maths in english". I am studying mathematics in Germany (I am from Germany, so english is not my native language) and have recently started ...
2
votes
2answers
199 views

Proof that the sequence $s_n = \frac{1}{n}$ converges to $0$

Prove that the sequence $s_n = \frac{1}{n}$ converges to $0$. I am writing this proof in order to help other people to understand better how to prove if a sequence converges and in particular why ...
1
vote
2answers
79 views

Book Recommendations for Writing Proofs

As an applied mathematics student coming from a small university, I have not had an adequate course in writing/formulating proofs for problems in advanced calculus/real analysis (my university has an ...
0
votes
1answer
52 views

Midpoint of a set as Mean? [closed]

Given a set of an odd number of terms: $x = \{a, b, c, ..., \}$ Consisting of $n$ elements. How is the midpoint of the set. A proof and explanation would be helpful: $$\frac{a + b + c + ... }{n} ...
0
votes
3answers
72 views

Isn't $\mathbb{P}$ already a probability measure, so what is there to prove?

Follow-up to Probability measure over finite sample space. This is a theorem from Casella and Berger's Statistical Inference: Let $S = \{s_1, \dots, s_n\}$ (sample space) be finite and $p_1, ...
1
vote
1answer
66 views

Proving a sequence is Cauchy (and convergent) by an infinite geometric sequence (something also with Lipschitz)

I've got a question concerning some weird kind of Lipschitz constant function, but it's an introduction course in Mathematics so Lipschitz hasn't passed the course yet. I should be able to prove this ...
1
vote
2answers
38 views

Prove Alternating Series Approximation

Prove if $S=\sum_{n=1}^{\infty}a_{n}$ is an alternating series with $\left | a_{n+1}\right | < \left | a_{n} \right |$, and $\lim_{n\to\infty}a_{n}=0$, then $\left |S-(a_{1}+a_{2}+\cdots+a_{n}) ...
0
votes
1answer
43 views

How to proove Hammer Split-graph Theorem?

Let $G=(V,E)$ be a Split Graph with $|V| \geq 4$. Then how to prove that: No induced sub-graph of G with 4 Vertices is a cycle with length 4 OR a pair of not incident edges? Well it must be from ...
2
votes
1answer
57 views

What are the requirements for a statement to have a constructive proof?

In general when trying to solve an excersise, or construct a proof, I always find myself looking at what strategy should I take to complete the proof. Many times I try to solve the excercise with a ...
0
votes
1answer
41 views

Prove that a tournament is irreducible if and only if it is strongly connected

If a graph is irreducible, by definition there will be no source or sink and it will be strongly connected. Is my proof above good and how do I prove the converse?
3
votes
1answer
62 views

No truth function that is expressed by a formula that uses only implication and equivalence connectives

I proved the following statement by induction: Let $A$ be a propositional formula which uses only the connectives $→$ and $↔$. Prove (by induction on the complexity of $A$) that if every ...
1
vote
1answer
43 views

If $\vec{v}$ is an eigenvector of $A$, then also $B\vec{v}$, when $AB = BA$ [duplicate]

I have the following problem: Let $V$ be a finite dimensional vector space. Let $A$, $B$ be linear maps of $V$ into itself. Assume that $AB = BA$. Show that if $\vec{v}$ is an eigenvector of $A$, ...