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
31 views

Help complete proof ('or' statement in conclusion): $a^2 \ge 7a \Rightarrow a\le0 \text { or } a\ge7$

Question Prove that if $a$ is a real number such that $a^2 \ge 7a$ then $a\le0 \text { or } a\ge7$ My Attempt We are given: $a \in \mathbb{R}$ $a^2 \ge 7a$ And need to prove: $a\le 0 \text{ ...
2
votes
1answer
70 views

Set difference between power sets

Show that if $A$ and $B$ are sets then $\varnothing \notin \mathcal P(A) − \mathcal P(B)$ . My Attempt: If $A$ and $B$ are sets, then $ \varnothing \subseteq A $ since the empty set is subset of ...
7
votes
4answers
1k views

What is the correct way of disproving a mathematical statement?

This question is motivated by my midterm exam. In this exam there was a question as follow: Question: If the following statement is true, prove it, otherwise disprove it. If $\mathbf{u}$ and ...
0
votes
1answer
64 views

question about the Darboux integral theorem proof

well, the sentence goes like this: Consider $f$ bounded function in $[a,b]$. $f$ is integrable IF AND ONLY IF $\forall\epsilon >0$ $\exists$ a partition $P$ of $\left[a,b\right]$ such that ...
1
vote
1answer
127 views

How to prove a very basic algorithm by induction

I just studied proofs by induction on a math book here and everything is neat and funny: the general strategy is to assume the LHS to be true, and use it to prove the RHS (for the inductive step). Now ...
2
votes
0answers
84 views

Simple Proofs in ZFC Set Theory

So I'll keep this real short and simple. In this document on page 23 there is a list of axioms. On page 24 there is a list of theorems that come from said axioms. I can prove them all except 3 and 5. ...
1
vote
1answer
62 views

Prove that $\dfrac{\sigma_1(n)}{n} = \sigma_{-1}(n)$ where $\sigma_x(n)$ is the sum of the $x$th powers of the positive divisors of $n$.

I computed $\dfrac{\sigma_1(n)}{n}$ and $\sigma_{-1}(n)$ on a good hundred values of $n$, and they seem to always match. For example: $\dfrac{\sigma_1(6)}{6} = \dfrac{1 + 2 + 3 + 6}{6} = ...
7
votes
4answers
733 views

The role of 'arbitrary' in proofs

Generally, when one is going to prove a result regarding a set of elements, they begin their proof with those first few pleasing words: "Suppose...is an arbitrary element in..." My question is, why ...
2
votes
2answers
113 views

Proof by induction that $(1^3 + 2^3 + 3^3+\cdots+n^3) = (1 + 2 + \cdots + n)^2$ [duplicate]

I'm sitting with the proof in front of me, but I do not understand it. $$A = \{n \in Z^{++} \mid (1^3 + 2^3 + 3^3+\cdots+n^3) = (1 + 2 + \cdots + n)^2\}$$ The first step of proof by induction is ...
1
vote
0answers
52 views

What is the relationship between division and proving an integer is odd?

I am trying to use proof by contradiction to prove: $101$ is an odd integer. I know that the first step is to assume that $101$ is even, so: $101 = 2q, q \in \mathbb{Z}$ Then I am stuck. I don't ...
2
votes
2answers
236 views

How to prove that the law of the excluded middle is necessary?

This is a follow up question to this answer by Carl Mummert to the question whether every proof with contradiction can also be proved without contradiction. As Carl Mummert pointed out, there are ...
1
vote
1answer
45 views

Prove that every subgraph of forest has at least one vertex of degree < 2

So I know that a forest is a graph that has no cycles. This is what I had in mind: Assume that we have the subgraph T, which has two options: to be connected or not. if it's connected it has to be a ...
3
votes
2answers
28 views

How do I eliminate mod from an expression?

If I have an expression such as $$ x = ((a \bmod b) - s) \bmod t, \quad 0 < a < b $$ And I want to step to $$ x = (a - s) \bmod t $$ Is acceptable to jump straight from the first expression to ...
4
votes
3answers
765 views

What is the relation A = B = C called in a proof?

When writing a proof if I have the relationship $$ A = B = C $$ And I want to use that to prove $$ A = C $$ I remember there being some term for it. What is that term, and what would be an ...
0
votes
2answers
44 views

Let $F$ be a field and $x, y\in F$. Prove:

Use field axioms to prove: a) $(−1) · (−x) = x $ b) If $x · y = 0$ then $x = 0$ or $y = 0$ I don't understand how to approach these questions. Does the field include $1$ and $0$ as well?
0
votes
1answer
65 views

Winning or Non-losing strategy for A or B

Find a winning or a non-losing strategy for the following game: Consider $25$ sticks arranged in a $5$ x $5$ square. Players alternately take any number of sticks from a single row or column. At ...
2
votes
1answer
38 views

$\frac{|| \overline{AM}||}{|| \overline{AB}||}=\frac{|| \overline{AN}||}{|| \overline{AC}||}=\frac{|| \overline{MN}||}{|| \overline{BC}||}$

$\Delta ABC$ is a triangle, $M$ is a point in the segment $\overrightarrow{AB}$ and $N$ is a point in the segment $\overrightarrow{AC}$, such that $\overrightarrow{MN}$ is parallel to ...
1
vote
3answers
81 views

Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$ [duplicate]

Use induction to prove that $2^n$ divides $(2n)!$ for any $n\in\mathbb{Z}_{\geq0}.$ How would you do the inductive step for this proof? I have the base case done.
1
vote
5answers
91 views

Matrix Power Formula

Prove that for a fixed $a \in \mathbb{R}$ we have the matrix power formula for all $n \in \mathbb{Z}_+$: $$\begin{pmatrix}a & 1\\0 & a\end{pmatrix}^n = \begin{pmatrix}a^n & ...
2
votes
1answer
46 views

Question about $e^T$ where T is a transformation

First off, I'm given a matrix $A$ s.t. the characteristic polynomial of $A$ is $p(a) = (-1)^nx^n+x^2-x+2$ and am asked to find $det(A^k)$ for a natural $k$ and $det(e^A)$. So from the polynomial I get ...
0
votes
4answers
185 views

Singleton sets are a subset

I am tasked with prove the following elementary result. I am concerned about being rigours enough in my proof: $$a\in S \iff \{a\} \subseteq S $$ My Attempt: Suppose $\{a\} \subseteq S $ . Then ...
3
votes
1answer
62 views

Do I need to use induction to have sufficient rigor in this proof?

I'm taking my first analysis class this summer. The professor asked us to prove that $a^{2n}-b^{2n}$ is divisible by $a+b$. After dorking around with the first couple of $n$ I was able to come up ...
3
votes
3answers
103 views

Beginner Proof about Primes

I am interested in understand the proof of infinitely many primes. It seems like quite an easy proof, ( I know there are many but I am referring to the proof that goes as follows); " Suppose there ...
0
votes
2answers
337 views

Prove that if $n^2 - 1$ is divisible by a prime number $p$ such that $n - 1$ is not divisible by $p$, then $n + 1$ is also divisible by $p$.

If this proposition is false, please give at least $3$ counter-examples, and try to modify the proposition so that it becomes true. If the proposition is true, please try to prove this even more ...
1
vote
5answers
141 views

If $a, b$ are relatively prime proof.

Prove that if $a$ and $b$ are relatively prime integers and $ab$ is a perfect square so are $a$ and $b$. Show by counterexample that the relatively prime condition is necessary. I dont know how ...
0
votes
1answer
30 views

How do I derive a contradiction from an assumption that is “not asymmetric”

Let $R$ be a transitive relation on a set $A$. Define another relation, $S$, such that, for any $x,y \in A$, $Sxy$ iff $Ryx$. Moreover, let $S$ be irreflexive. Prove: $S$ is asymmetric on $A$. ...
2
votes
3answers
139 views

Metric Space, Induced topology

I'm trying to show that the metric topology is indeed a topology. To do so, I want to show the following three statements are true: $\emptyset$ and X are in $\tau$ finite intersections of open sets ...
1
vote
2answers
107 views

Projections are open maps. Why might I be wrong?

I got this problem from Munkres, my idea is similar, but comparing to the actual solution, I missed at least 4 steps. Prove that the projection maps $\pi_1 : X \times Y \to X$ and $\pi_2 : X ...
4
votes
2answers
116 views

Is there a “rule of thumb” of what can be reasonably omitted from a proof at the graduate student level?

As someone entering graduate school this fall, this is something I would like to know. For undergraduates, I know that professors generally want students to show very rigorously and clearly their ...
0
votes
2answers
77 views

How would this problem need the Mean Value Theorem?

I'm asked to square the inequality and use the Mean Value Theorem to prove that $$\sqrt{1+x} < 1 + \frac{x}{2}$$ for $x>0$. Unfortunately, I don't really understand why I would need such a ...
2
votes
2answers
70 views

Free and bound variables in “if” statements

The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion $\forall x(x>2)$ is false. In the first case ...
2
votes
1answer
47 views

Proving a theorem on limits

I need to prove that: If $$\lim_{(x,y) \to (a,b)}f(x,y)= 0 \text{ and } g(x,y)\leq k,$$ then: $$\lim_{(x,y) \to (a,b)}f(x,y) g(x,y) =0.$$ My approach is like follow: ...
5
votes
2answers
95 views

Idea of a proof by contradiction

Is the idea of a proof by contradiction to prove that the desired conclusion is both true and false or can it be any derived statement that is true and false (not necessarily relating to the ...
0
votes
0answers
63 views

complete logic for proving inequalities

Last semester I took a course on algorithm analysis a big part of which was proving that the running time function of a program was in the set $O(f(x))$ for some $f$. To prove $f\in O(g(x))$ one ...
1
vote
2answers
53 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
4
votes
0answers
309 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $\mathscr{C}^1$ on $(c,d)$. Then ...
4
votes
3answers
131 views

Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$

Let $S\subset\mathbb{R}$ be a non-empty bounded above set. Then there exists a monotone increasing sequence $\{x_n\}\subset S$ such that $$\lim_{n\to\infty}x_n=\sup S.$$ I'm struggling with ...
1
vote
2answers
148 views

Is there a specific name for a corollary of a conjecture?

How do you call a corollary of a conjecture? Is there a specific name for it? Can it be called simply 'corollary'? Can't it be called 'corollary'? I mean, does the label 'corollary' imply that the ...
1
vote
0answers
53 views

Proof Validation Function From Integers to Rationals is Continuous

I am teaching myself real analysis, so any help is greatly appreciated. Let the function be defined as $F : Z \rightarrow Q$ where $Z$ is the set of integers and $Q$ is the set of rational numbers, ...
2
votes
0answers
63 views

Velleman's How to prove it. Partial order proof.

Theorem: Suppose that $R$ is a partial order on $A$, $B_1 ⊆ A$, $B_2 ⊆ A$, $x_1$ is the least upper bound of $B_1$, and $x_2$ is the least upper bound of $B_2$. Prove that if $B_1 ⊆ B_2$ then ...
0
votes
2answers
125 views

Suppose $F$ and $G$ are families of sets.

Suppose $F$ and $G$ are families of sets. Prove that $\bigcup F$ and $\bigcup G$ and are disjoint iff for all $A∈F$ and $B∈G$ , $A$ and $B$ are disjoint. It has been suggested to use contrapositive ...
1
vote
2answers
43 views

Proofs involving Well-Defined and One-to-One

Chartrand, 3rd Ed, P224-225: Define a relation $R$ as a relation from A to B. $R$ is well-defined means: $(a,b), (a,c) \in R \implies b = c$. P220: A function $f: A \to B$ is one-to-one ...
3
votes
1answer
60 views

$f'$ strictly increases and $f'(c)=0$. There exist $x_1 < c < x_2$ such that $f'(c)=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$

Question: Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Assume that $f'$ is strictly increasing. Show that for any $c\in(a,b)$ such that $f'(c)=0$, there exist $x_1, x_2 \in [a,b], ...
3
votes
0answers
91 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
2
votes
2answers
45 views

Prove by induction that $\sum_{i=1}^n i!\times i=(n+1)!-1$ for all $n\in \mathbb{N}$

So far I have, If $P(n):\sum_{i=1}^n i!\times i=(n+1)!-1$, then $P(1):\sum_{i=1}^1 i!\times i=1$ and $(1+1)!-1=1$ , so P(1) is true. I know I now have to assume P(K) is true, such that ...
0
votes
2answers
40 views

I know the limit of a subsequence exists, but how do I find it?

I know intuitively that for: $(a_n) = (1,1,2,1,2,3,1,2,3,4, ...)$ for any $m$ in the positive natural number there is a subsequence (m,m,m,..) that tends to m as n tends to infinity. I believe I ...
2
votes
3answers
61 views

Strong Induction, assuming k<n where k and n are not numbers

In strong Induction for the induction hypothesis you assume for all K, p(k) for k If for example I am working with trees and not natural numbers can I still use this style of proof? For example if I ...
2
votes
1answer
209 views

Proof involving maximum weight of edge in minimum spanning tree

Let $G$ be a minimum spanning tree of a complete graph. Let $e$ be the maximum weight edge in $G$. I'd like to proof that given any other spanning tree $G'$ of this graph, being $j$ the maximum weight ...
1
vote
1answer
32 views

How to prove a bijection?

I know what a bijection is and how to prove it when given a function, but how to do it when you are only given sets.
3
votes
1answer
72 views

$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$.

Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b). $ If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$. I tried doing it directly but ...