0
votes
2answers
28 views

Proving the arithmetic mean equals the geometric mean when $a=b$.

Arithmetic mean $a,b \in \mathbb R$ is $A(a,b)=\frac{a+b}{2}$ Geomtric mean $a,b \in\left[0,\infty\right]$ is $G(a,b)=\sqrt{ab}$ I'm trying to prove that $G(a,b)=A(a,b)$ if and only if $a=b$. ...
2
votes
1answer
60 views

Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+…+x+1)$

So what I am trying to prove is for any real number x and natural number n, prove $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$$ I think that to prove this I should use induction, however I am a bit stuck ...
0
votes
0answers
36 views

Proof Strategy: Induction Summation of Series

Let $P(n)$ be the following statement: $$\sum\limits^{n}_{i=0}r^i = \dfrac{1-r^{1+n}}{1-r}\text{ for all }n \in \mathbb{N}\text{.}$$ I am stuck at the base case: $$P(1):1 + r = ...
0
votes
1answer
25 views

Prove that for any $\{x_{m_k} \}\in I_n$, where $I_n$ are dyadic intervals, $\lim_{n \to \infty} x_{m_k} =c$

Proving for any $\{x_{m_k} \}\in I_n$ that:$$\lim_{n \to \infty}\{x_{m_k} \}=c$$ I have been trying to solve this problem, but i dont know how to write it properly, so i need your help whit ...
2
votes
0answers
34 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
0
votes
0answers
31 views

Differentiability and $L^1, L^2$ spaces

If $f\in L^1(\mathbb{R})$ then $\frac{d}{dx}\{f(x)\}\in L^1(\mathbb{R})$ where we have given that $f$ is of compact support.
0
votes
3answers
25 views

A property regarding intervals

While I was solving a problem on TopCoder I used the following assumption. I have n intervals: $ [a_1,b_1], [a_2,b_2],...,[a_n,b_n]$ and a number $T$ such that: $$ a_1 + a_2 + ... + a_n \leq T \leq ...
1
vote
0answers
8 views

Finding posterior of normal distributions and logistic regression.

$P(w_0 | x) = \frac{1}{1 + e^{-log\frac{P(x|w_0)}{P(x|w_1)}-log\frac{P(w_0)}{P(w_1)}}}$ Note: x = $[x_1, \dots, x_d]^T$; a $d$ dimensional vector. $w$ can take on one of two values: $w_0$ or $w_1$. ...
0
votes
1answer
70 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
1
vote
2answers
27 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
0
votes
2answers
24 views

Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...
2
votes
3answers
34 views

Proof involving lcm and biconditional statement.

Suppose $a,b\in\mathbb{Z}$. Then $a = \operatorname{lcm}(a,b)$ if and only if $b\mid a$ Unsure of how to approach this problem.
0
votes
3answers
34 views

Proof for a non-conditional statement

I'm having a bit of trouble doing this proof. If $a\in\mathbb{Z}$, then $a^3 \equiv a \pmod 3$. I know how to do proofs if there were conditional statements but not sure how to prove this with ...
1
vote
0answers
59 views

Prove: $a^m\cdot a^n \cdot a^p=a^{m+n+p}$

How can I prove the following: Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove: 1) $a^0\cdot a^0 \cdot ...
1
vote
2answers
39 views

Invertible Proof with transposed matrices

Let A, B, C be square matrices that are invertible. Say I want to express X with no inverses Say $$ (A^{T}A)^{-1}(X +B^ {T})(C^{-1}B^{-1})^{T} = I. $$ I know that $A^{T}A$ = $I$, but where can I go ...
0
votes
3answers
29 views

Help to prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$

Prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$ My attempt: $\begin{align} (x,y) \in (A \times B) \cup (C \times D) & \Rightarrow & (x,y) \in (A \times B) \vee ...
0
votes
2answers
27 views

Proof for combination using a specific definition

Suppose $n,k,\in\mathbb{Z}$ and $0\leq k \leq n $ prove using the following definition: if n and k are integers then $\binom{n}{k}$ denotes the number of subsets that can be made by choosing k ...
1
vote
1answer
33 views

Proof that P is an Orthogonal Projection

I'm studying linear algebra using Axler's text and am stuck on 6.17. The problem is: Prove that if $P \in \mathcal{L}(V)$ is such that $P^2 = P$ and every vector in $\operatorname{null} P$ is ...
0
votes
3answers
92 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
2
votes
3answers
164 views

Proof that $\int \frac{1}{x}$ is $\ln(x)$

When I was learning Calculus AB and Calculus II/III at my high school, I noticed that our textbooks never gave a full fundamental proof that $\int \frac{1}{x}$ is $\ln(x)$, and rather said that when ...
30
votes
5answers
3k views

Is it okay to reverse engineer proofs in homework questions?

In a linear algebra text book, one homework question I received was: Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$. Where $\mathbf{a}$ and ...
1
vote
1answer
60 views

Proof by contradiction: logarithm

I need to prove by contradiction that $\log_2(3)$ is irrational. I'm really unfamiliar with logs to be honest, it's been awhile since I've done them and I'm unsure of how to approach this. Any help ...
0
votes
3answers
43 views

Proof by contradiction proving both numbers are not odd.

I have to do a proof by contradiction: Suppose $a,b,\in\mathbb{Z}$. If $4| (a^2 + b^2)$ then a and b are not both odd. So far I know that I need to prove that if $4|(a^2+b^2)$ then a and b are both ...
3
votes
2answers
72 views

Prove $A = (A \setminus B) \cup (A \cap B)$

Prove $A = (A \setminus B) \cup (A \cap B)$ Logically, this is clearly true. I can explain why: start with $A$, remove all elements in $B$ and then add in any elements in both $A$ and $B$, which ...
4
votes
0answers
97 views

Saddle Points on Matrices

Let $n$, $m$ be positive integers. Suppose that $A$ is a $2$ x $n$ or an $m$ x $2$ matrix and that it has a saddle point. Show that among the saddle points of $A$ there exists at least one which ...
0
votes
0answers
30 views

Matrix Saddle Points and Dominance

I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it can be used in games. I wanted to know how to prove a problem like ...
0
votes
1answer
33 views

Proof regarding division with a remainder

Let $a\in\mathbb{Z},n\in\mathbb{N}$. If $a$ has a remainder $r$ when divided by $n$, then $a\equiv r\pmod n$ I've done some of these questions before with modulus and division, but I'm unsure of how ...
2
votes
5answers
69 views

Proof regarding divisibility

if $n\in\mathbb{Z}$, then $4$ does not divide $(n^2 - 3)$ I'm not sure how to approach this question, I know how to do questions that involve proving that it does divide but I'm unsure of how to do ...
0
votes
2answers
24 views

Modulus related proof help

I need to prove this via either direct proof, or contrapositive. Unsure of the best way to approach this. if $a \equiv b\mod n$ and $c \equiv d\mod n$, then $ac \equiv bd\mod n$ So far I have: ...
2
votes
1answer
39 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
587 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 ...
6
votes
4answers
656 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 ...
1
vote
0answers
40 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
100 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
23 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 ...
0
votes
1answer
52 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
32 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
62 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
48 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 & ...
0
votes
4answers
147 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 ...
2
votes
3answers
77 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
62 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 ...
0
votes
0answers
26 views

U-substitution proof by 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 $C^1$ on $(c,d)$. Then define ...
2
votes
1answer
44 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: ...
0
votes
0answers
19 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
41 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
74 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 $C^1$ on $(c,d)$. Then define ...
2
votes
0answers
39 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
50 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 ...
2
votes
2answers
33 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 ...