# Tagged Questions

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### 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 $a=b$. ...
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### 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 ...
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### 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 $w_1$. ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\frac{|| \overline{AM}||}{|| \overline{AB}||}=\frac{|| \overline{AN}||}{|| \overline{AC}||}=\frac{|| \overline{MN}||}{|| \overline{BC}||}$

$\Delta ABC$ 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 ...
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### 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.
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### 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 ...
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### 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 ...
### 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 ...
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 ...