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
4answers
227 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
84 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
27 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
31 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
65 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
100 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
42 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
31 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
55 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
36 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 ...
0
votes
0answers
48 views

derivative $1 \over x$ -proof

proving $\frac{1}{x}$ by definition $$(\frac{1}{x})'=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} {\frac{-h}{(x+h)x}\over h}=lim ...
0
votes
2answers
37 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
42 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
56 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
25 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
33 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
35 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 ...
1
vote
3answers
86 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
17 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
34 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
23 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
35 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 ...
-4
votes
1answer
117 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
48 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 ...
0
votes
2answers
44 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 ∈ ...
0
votes
0answers
30 views

Determining if two bounds are true

Question says assume $f$ and $g$ have a domain of the integers, and target space of the real numbers. $f$ and $g$ are bounded. Prove if the following statements are true or give a counterexample: if ...
1
vote
1answer
23 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
39 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
52 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
80 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
81 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
3answers
81 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 - ...
10
votes
4answers
599 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
189 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
46 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
43 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
64 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
27 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
29 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
29 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
50 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
23 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
55 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
35 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 $A$, ...
2
votes
1answer
31 views

Differentiability implies continuity - A question about the proof

I have a question, to aid my understanding, about the proof that differentiabiility implies continutity. Differentiability Definition When we say a function is differentiable at $x_0$, we mean that ...
1
vote
1answer
58 views

Proving by induction on the length of a propositional formula?

I'm having a little trouble understanding the following proof question because I'm unsure what defines the 'length' of a propositional formula, I've seen multiple definitions whether it's the number ...
1
vote
2answers
54 views

How to prove the following inequality: $\frac{\sqrt{n + 1}}{\sqrt{n}} - 1 \leq \frac{1}{2n - 1}$

As a part of my practice for an upcoming mid-term, I managed to simplify the following inequality to what you see here: $$\frac{\sqrt{n + 1}}{\sqrt{n}} - 1 \leq \frac{1}{2n - 1}$$ And honestly I'm ...
0
votes
2answers
35 views

How to formally prove that an element belongs to a sequence of sets.

Take any $\delta \in [ \frac{1}{2}, 1)$, I want to show that there always exists an $n$ s.t. $\delta \in [\frac{1}{2}, 1 - \frac{1}{n}) $. Can one obtain an explicit relationship between $\delta$ and ...
4
votes
0answers
62 views

How to Prove It, Exercise 6.5.9

Suppose $R$ is a relation on $A$ and $S$ is the transitive closure of $R$. If $(a, b) \in S$, then there is some positive integer $n$ such that $(a, b) \in R^n$, and therefore by the well-ordering ...
2
votes
0answers
33 views

Clever proofs that prove one identity is equal to another, without going through the original identity?

In a previous question I attempted to formalize the argument of going from one proof of an identity to another, which turned out to be harder than I thought. The thing is, while it may be impossible ...