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

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1answer
40 views

Calculational proofs and Natural Deduction system.

We are all familiar with calculational reasoning. A proof by calculational reasoning basically proceeds by forming a chain of intermediate results that are meant to be composed by basic principles, ...
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1answer
170 views

Proof of existence of “peaks” in The Sequential Compactness Theorem

According to Bolzano–Weierstrass theorem: How do we guarantee that EVERY sequence of real numbers has either infinitely or finitely many "peaks"? (Note that a subsequent is ordered in a way of ...
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2answers
44 views

Proving or Disproving statements using sets

I just don't seem to get proofs or set theory so hopefully my question makes sense. I'm not sure when I should or shouldn't use an example to prove or disprove a statement? One example question is, ...
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2answers
119 views

Prove using PMI that if $A$ is denumerable and $B$ is finite, then $A \cup B$ is denumerable.

This is what I have thus far: Claim: If $A$ is denumerable and $B$ is finite, then $A \cup B$ is denumerable. Proof. Suppose $A$ is denumerable and $B$ has $n$ elements and $B = \{b_1, b_2, b_3, ...
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4answers
93 views

Proving that $x^2 + 4$ is not divisible by $3$

I need to show the following: For any integer $x$, $x^2 + 4$ is not divisible by $3$. I was trying proof by contraposition, but I do not believe that is the most efficient way to go about this. ...
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0answers
23 views

Looking for references for learning the words and sentences used in proofs

I'm familiar with textbooks on logic, proof techniques, and sets. But I have yet to encounter a textbook that dives into the language used w/ definitions and sentence structure used in proofs, for ...
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4answers
283 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 ...
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63 views

Is my proof of the Division Algorithm 'enough'?

Recently when learning number theory I was introduced to the proof of the division algorithm, it can be found here http://www.oxfordmathcenter.com/drupal7/node/479. However, I decided to prove it ...
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1answer
66 views

Why aren't definitions well formed formulas?

Why aren't definitions well formed formulas? For instance, the definition of an additive inverse is: "Let $x \in \Bbb Z$. Then the additive inverse of $x$ is $y \in \Bbb Z$ such that $x+y=0$". Why ...
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2answers
38 views

Proofs with for all statements including uniqueness and divides [closed]

Let $\mathcal{A}$ be a nonempty finite set of positive integers, with $\forall$ r $\in$ $\mathcal{A}$, $\forall$ s $\in$ $\mathcal{A}$ : r|s or s|r. (i). Prove $\exists$t $\in$ $\mathcal{A}$: t|a, ...
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2answers
56 views

Delta epsilon argument in general

When I want to prove something in mathematics fe an expression goes to zero, I can either use basic rules of 'limits' or I can use the epsilon-delta method. I have a feeling that it's more consistent ...
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2answers
123 views

What exactly is the 'induction trap'

I've looked everywhere, and I've looked at a lot of examples. I don't quite understand what about the induction trap is so wrong. The most common example is the graph theory tree example (page 5 here: ...
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1answer
42 views

Is it necessary to prove equality from both sides?

I have asked this question yesterday, and my friend told me, to rememeber to "prove it" also from the other side e.g. Let x $\in$ Conv($M+u$).....then $x$ $\in$ Conv($M$)+ $u$. Why would somebody ...
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3answers
73 views

How does logic and elementary set theory work together to prove $A \cup \varnothing = A$?

In How do I prove $A \cup\varnothing = A$ and $A \cap\varnothing = \varnothing$ A proof was given reproduced here: Prove: $A \cup \varnothing = A$ Let $a\in A\cup \varnothing$. Then $a\in A$ or ...
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1answer
33 views

Correctness of Idea of Big O Proof

I have this big O proof and was wondering about the correctness of my rough work. Could anyone confirm if my idea for my proof is correct? Here is the question: Let ...
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1answer
25 views

Proving/disproving statements with a given context of natural numbers.

How do I prove the following statements or their negations in the context where $x$ and $y$ are rational numbers in the closed interval $[-\sqrt{2}, \sqrt{2}]$? Statement 1: $\forall x \exists y\; x ...
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2answers
827 views

Help with graph induction proof

I'm trying to prove : Given a simple graph $G$ with $n$ vertices, where $n$ is even, prove that if every vertex has degree $\dfrac n2 + 1$, then $G$ must contain a (simple) $3$-cycle. A (simple) ...
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1answer
30 views

Proving $m^*(A \cup B) \leq m^*(A) + m^*(B)$

Let $m^*(A) = \inf\{\sum\limits_{k=1}^\infty {E_k}: \{E_k\} \text{ is a cover of } A\}$ where each $E_k$ is an interval (i.e. continuous open set) Prove: $m^*(A \cup B) \leq m^*(A) + m^*(B)$ ...
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0answers
10 views

Find the volume of h(P) in terms of volume of P

Where h:$\mathbb R^n \rightarrow R^n$ is the function h(x) = $\lambda x$ and P is a k-dimensional parallelopiped in $\mathbb R^n$ Attempt at the solution : Let $x_1,....,x_k$ be vectors in $R^n$ ...
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2answers
104 views

Is it good to use mean value theorem in $\epsilon-\delta$ continuity proofs?

I wanted to prove $f(x) = \cos(x)$ is continuous using $\epsilon-\delta$ proof Couple of posts on MSE appealed to MVT to resolve this problem. Namely: $\exists c \in [x,x_o]$ s.t. ...
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0answers
37 views

Help on Big O proof

I need some help with a big O proof. I think I have a proof but I feel like some of the steps aren't logically compatible. The Question: For all functions f,g with domain $\mathbb{N}$ that maps to ...
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1answer
74 views

Let $1 + 2^m = 3^n$. What the max value of $(m+n)$?

How do I determine the maximum value of $(m+n)$ if $m$ and $n$ are natural numbers if $1 + 2^m = 3^n$ holds? I have got $\text {max} (m+n)$ to be $5$ so far, but I do not know how to determine whether ...
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0answers
34 views

How to generate/validate unique fractal?

There are many known fractals that exist such as Mandelbrot, Cantor set, or the Koch curve, Sierpinski Triangle. What I am curious about, is how one could go about creating their own, unique fractal ...
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0answers
42 views

Let $A,B,C$ be sets such that $\mathcal{P}(A) \cup \mathcal{P}(B) = \mathcal{P}(C)$. Then either $A=C$ or $B=C$.

Let $A,B,C$ be sets such that $\mathcal{P}(A) \cup \mathcal{P}(B) = \mathcal{P}(C)$. Then either $A=C$ or $B=C$. I just need a little direction with this. I know that since $\mathcal{P}(A) \cup ...
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1answer
26 views

Let $A,B \subseteq X$. If $A \subseteq B$, then $(X\setminus B) \subseteq (X\setminus A)$

Let $A,B \subseteq X$. If $A \subseteq B$, then $(X\setminus B) \subseteq (X\setminus A)$. Using a sort of diagram I can easily convince myself this is true. I assume I must use a proof by ...
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1answer
40 views

Proof by contradiction to this inequality

Prove, by contradiction, that if $w$, $z$ $\epsilon$ $\mathbb{C}$ such that $|w|$ $\leq 1$ and $$w^{n} z + w^{n - 1} z^2 + \cdots+ wz^n = 1$$ then $|z|$ $\gt$ $\frac{1}{2}$ I have been thinking ...
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2answers
43 views

Question about 'strong' assumptions and proving 'strong' result.

When someone says "we can prove a stronger result", it means the new statement is more general and better than the previous. However, when we add conditions to an argument, add more assumptions, we ...
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2answers
35 views

True or false: the contrapositive statement of “$\text{not } A\implies \text{not } B$” is “$A\implies B$”?

Question : State whether 'True' or 'False' and justify your answer giving reason. The contrapositive statement of "$\text{not } A\implies \text{not } B$" is "$A\implies B$" where $A$ and $B$ ...
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1answer
53 views

Why does $\lim_{x \to \infty} \frac{n^x + x^2}{n^x +x} = 1$ for a constant $n \gt 1$?

Why does $\lim_{x \to \infty} \frac{n^x + x^2}{n^x +x} = 1$ for a constant $n \gt 1$ but infinity when constant $n \leq 1$ ? I understand intuitively that when $n$ is less than or equal to $1$, $n^x$ ...
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1answer
74 views

What words do you use to describe a bad proof?

I was writing up a proof, but wasn't satisfied with what I was coming up with. The logic is there, but I wasn't able to express it clearly in math lingo. I was about to describe my work as "floozy" ...
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1answer
67 views

What is the most basic way to show that $\emptyset \in S$

Let $S$ be a set, what is the most basic way to show that $\emptyset \in S$? I am asking because sometimes a question in involving a topology $\tau$ or a $\sigma$-algebra $\Sigma$ will want you to ...
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4answers
5k views

Proving the so-called “Well Ordering Principle”

Is there anything wrong with the following proof? Theorem. Every non-empty subset $B \subset\mathbb{N}$ has a least member. Proof. Assume not. Then, of necessity, we'd have to have $B=\varnothing$, ...
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1answer
48 views

How can I prove that the centers of three overlapping squares and the shared point form a parallelogram?

Consider a square. Let $P$ be any point on the base of the square. Connect each of the upper vertices to this point and use these segments as sides to create two more squares (outward). The picture ...
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0answers
20 views

Computing $\alpha^*w$ in general

Let $A$ be open in $R^k$, let $\alpha : A \rightarrow R^n$ be of class $C^\infty$. Let $x$ denote the general point of $R^k$, let $y$ denote the general point of $R^n$ If $I = (i_i, ...,i_l)$ is an ...
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0answers
15 views

Show that $\alpha^*(dy_i) = d(\alpha^*y_i) $

where $y_i:\mathbb R^n \rightarrow R$ is the $i^{th}$ projection function in $\mathbb R^n$ Given: Let A be open in $R^k$; let $\alpha: A \rightarrow R^n$ be a $C^{\infty}$ map. Let x denote the ...
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2answers
94 views

need help proving an interval

I am trying to proof $$\frac {1} {ek} \le \frac {1}{k} (1 - \frac {1}{k} )^{k-1} \le \frac {1}{2k} $$ for k>=2 to prove this I first multiply by k getting $$\frac {1} {e} \le \left(1 - \frac ...
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1answer
49 views

Proof of a Four-Pole Tower of Hanoi

Four-Pole Tower of Hanoi: Suppose that the Tower of Hanoi problem has four poles in a row instead of three. Disks can be transferred one by one from one pole to any other pole, but at no time may a ...
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0answers
28 views

If $|ax^2+bx+c|\le1$ for $|x|\le1$(redundant?), then prove that $|cx^2+bx+a|\le2$ for $|x|\le1$. [duplicate]

If $|ax^2+bx+c|\le1$ for $|x|\le1$, then prove that $|cx^2+bx+a|\le2$ for $|x|\le1$. $a,b,c$ are real. I discovered this to be a duplicate of $|px^2+qx+r|\le1$ for all $x$ in $[-1; 1]$, show that ...
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3answers
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prove the Limit of the sequence $A_{n}$, when $A_{1} = \sqrt{3}$ and $A_{n+1} = \sqrt{3A_{n}}$

is it sufficient to show that since the limit of a sequence $A_{m}$ where $\forall m, \;A_{m}=\sqrt{3}$ is just $\sqrt{3}=3^{\frac{1}{2}}$, the limit of $A_{n}$ will be 3 to some power? I know from ...
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1answer
39 views

Show that monotonicity directly follows countable additivity

In these notes http://math.gmu.edu/~dwalnut/teach/Math776/Spring11/776s11lec03_notes.pdf it says countable additivity implies monotonicity of measure I want to show that $\mu(\bigcup_{k = 1}^\infty ...
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2answers
56 views

Prove that $7^{\frac 14}$ is not rational using the Unique Factorization Theorem.

I am currently trying to prove this using the Unique Factorization Theorem and I am stuck. I attempt to prove this BWOC and assume $7^{\frac 14}$ is rational so that it can be expressed as $\frac ...
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2answers
37 views

Let $G$ be a group and $H_{1},H_{2}\leq G $. Then $H_1 \cup H_2 \leq G \iff H_1 \subset H_2$ or $H_2 \subset H_1$

Let $G$ be a group, $H_{1},H_{1} \leq G $. Then $H_1 \cup H_2 \leq G \iff H_1 \subset H_2$ or $H_2 \subset H_1$ I'm stucked at this very trivial proof of groups. Here's my attempt: $(\Leftarrow)$ ...
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1answer
25 views

if $d$ is the greatest common divisor of $a, b \in \mathbb{Z}$ show that $(a) + (b) = (d)$

I've got the basic idea of this proof just not sure how to write it formally. If $d = gcd(a, b) $, then $ d = au + bv $, where $a$ and $b$ are not both zero. $(d)$ is the set of all multiplies of ...
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2answers
98 views

Writing proof (disproof) prime number formula

Please prove or disprove: If $n \in ℤ^{+}$, then $n^{2} + n + 41$ is prime. I know that the above statement is not true because if you plug in 41 for $n$, the result is not a prime number. How can I ...
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1answer
35 views

If R is a commutative ring, but has no identity is c and element of I

Problem statement: Let $c \in R$ and let $ I = \{rc\mathrel{|}r\in R\} $ If R is a commutative but has no identity, is c an element of the ideal I. Proof: Suppose $c \notin I$ and R is a ...
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3answers
43 views

Show that there are infinitely many positive integers A such that 2A is a square, 3A is a cube and 5A is a fifth power.

Show that there are infinitely many positive integers A such that 2A is a square, 3A is a cube and 5A is a fifth power. Using some arithmetic, I felt that if $A = 2^{15k}3^{20k}5^{24k}$ then it ...
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0answers
22 views

Proving Infinitely Ascending Chain of Subobjects

How would you prove that an infinitely ascending chain of subobjects of an object $X$ in $\mathcal{C}$ is stationary given that only finitely many preorders in the chain that are not isomorphisms in ...
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2answers
184 views

How to study for hard math proofs?

Most of the content is new to me and there are a lot of theorems and proofs that I am learning; not that I need to know all of them but I enjoy to learn more. Some of the concepts (like open sets) or ...
4
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1answer
46 views

Show $\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$

I am trying to give a non-algebraic proof for this equality: $$\dbinom{a}{b}\dbinom{b}{c}=\dbinom{a}{a-c}\dbinom{a-c}{a-b}$$ So far, I could only use the identity $\dbinom{x}{y}=\dbinom{x}{x-y}$. ...
0
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0answers
26 views

Experimenting With Algebraic Abstraction

I am a novice attempting to understand the "abstract" components of mathematics. Consequently, I have devised mathematical statements that I would like to be verified by knowledgeable users using the ...