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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0
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3answers
41 views

How to write $a|b \wedge ((a=0 \wedge b=0) \vee (a\ne0 \wedge b\ne0))$

How can I write the expression $$a|b \wedge ((a=0 \wedge b=0) \vee (a\ne0 \wedge b\ne0))$$ concisely and clearly in English? A direct translation yields $a$ divides $b$ and either {$a$ and $b$ ...
1
vote
1answer
26 views

To prove these sets are equal without using modulo arithmetic.

Prove $\{3t : t \in \mathbb Z\} \cup \{3t + 1 : t \in \mathbb Z\} \cup \{3t + 2 : t\in \mathbb Z\} = \mathbb Z.$
1
vote
1answer
67 views

Odd town Even town explanation.

I am struggling to understand the solution to the following problem: If $\mathcal F\subset 2^{[n]}$ such that for each $F_1$ and $F_2$ in $\mathcal F$ we have $|F_1|,|F_2|\equiv 1 \bmod 2$ and ...
0
votes
1answer
33 views

How prove that $|QA| < |QC|$ in triangle?

$ABC$ is a triangle with a right angle at $A$, and $|AB|$ > $|AC|$. The point $D$ is defined so that $BCD$ is equlateral and $AD$ intersects $BC$ at $P$. The point $Q$ is defined so that $QDP$ is ...
1
vote
1answer
34 views

For what k is $\mathcal{M}_{m \times n}$ isomorphic to $\mathbb{R}^{k}$?

For what k is $\mathcal{M}_{m \times n}$ isomorphic to $\mathbb{R}^{k}$ ? I get a feel but am unable to prove it.
4
votes
1answer
78 views

How show that $ABC$ is equilateral?

Let $D$, $E$ and $F$ be three points on sides $BC$,$AC$ and $AB$ of triangle $ABC$ such that lines $AD$, $BE$ and $CF$ concur at point $M$. If three trianles $MDB$, $MCE$ and $MAF$ have equal areas ...
3
votes
1answer
19 views

Proof About Point and Triangles

Suppose we are given $n$ points in a plane, where $n\ge 4$ and no 3 of the points are collinear. If $k$ distinct triangles are designated with vertices among the $n$ points, show that no more than ...
1
vote
1answer
44 views

How prove that $AD>BE$ in triangle?

Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD>BC$ . The point $E$ on $CA$ is defined by the equation $\frac{AE}{EC}=\frac{BD}{AD-BC}$ .How prove that $AD>BE$?
0
votes
1answer
32 views

How to prove $|q|\ge 1 \Rightarrow |a|\ge |d|$?

Let $a,d,q \in \mathbb{Z}$ and $a=dq$ How do I show that $|q| \ge 1 \Rightarrow |a| \ge |d|$? I've tried: $|q|\ge 1 \Rightarrow (q>1 \text{, if } q>0) \text { or } (-q>1 \text{, if } ...
0
votes
2answers
60 views

Prove intersection of open balls is another open ball

I was wondering how I would prove that an intersection of two open balls is also another open ball. The definition I have of an open ball is: If x $\in X$ and $\epsilon > 0$, $B_{\epsilon}(x) :=$ ...
2
votes
4answers
73 views

Is this a valid proof for the existence of a rational number between any two real numbers?

Given $a, b \in \mathbb R$ with $a<b$, prove that there exists some $r \in \mathbb Q$ such that $a<r<b$. Before I prove the main statement, there's a lemma I'd like to prove: Lemma ...
0
votes
2answers
35 views

Trying to disprove a statement - some partial working included

I am trying to find a counter example to show that the statement below is false, but I am having difficulty in trying to find a reasonable argument. Here is the statement: $n^2-12n + 35 \geq 0$ for ...
0
votes
3answers
85 views

Prove $x^2=t$ for any $t>0$ [duplicate]

Prove for any positive number $t$, there is a solution for $x^2=t$. So we want to show that $x^2=t$ for $t\geq0$. We can break this into two cases: Case 1: Assume $t=0$, then we have $x^2=0$ ...
2
votes
2answers
57 views

Surjectivity of a piecewise function $f:(-1,1)\to \mathbb R$

Function $f$ is defined as $f: (-1, 1) \to \mathbb{R}$. $$ f(x) = \begin{cases} -x/(x-1),&x\geq 0 \\ x/(x+1),&x \leq 0 \end{cases} $$ Let $y \in \mathbb{R}$. How would I prove that there ...
4
votes
1answer
76 views

The “converse” of $P\rightarrow(Q\rightarrow R)$

As everyone know, even when reading mathematics books, a paragraph written in natural language contains much more information than just its purely logical translation. For my next tutor session, I am ...
0
votes
1answer
55 views

Prove that a function is continuous at x =0

I need to prove that $f$ continuous at $(x)=0$ using a $\epsilon$- proof $$ f(x) = \begin{cases} x/(1-x),&x\geq 0 \\ x/(1+x),&x \leq 0 \end{cases} $$ So this is what I have so far: Let ...
1
vote
3answers
44 views

Help with proof of contrapositive of well-ordering principle

Prove by induction on $n$ that if $A$ is a set of positive integers without a least element, then $\mathbb{N}_n \subseteq \mathbb{Z}^+ - A$ for every $n$ so that $A$ is the empty set. I don't ...
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.
2
votes
2answers
49 views

Proof of multiplicative inverse for polar complex numbers [duplicate]

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(\cos(\alpha)+i\sin(\alpha))$. I can do ...
0
votes
0answers
46 views

Prove that a defined function g is continuous for a certain point

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Let $c \in \mathbb{R}$. Define the function $g_{c}: \mathbb{Q} \to \mathbb{Q}$ via: ...
1
vote
3answers
45 views

Proof for complex numbers and square root

Use the polar form of complex numbers to show that every complex number $z\neq0$ has two square roots. I know the polar form is $z=r(\cos(\alpha)+i \sin(\alpha))$. I'm just not sure how to do this ...
1
vote
2answers
58 views

Use polar complex numbers to find multiplicative inverse

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(cos(\alpha))+i(sin(\alpha))$. I can do ...
2
votes
2answers
73 views

Prove that a function of the rational numbers $\mathbb{Q}$ with subspace topology inherited from $\mathbb{R}$ is injective

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Suppose $g: \mathbb{R} \to \mathbb{R}$ and $h: \mathbb{R} \to \mathbb{R}$ are ...
0
votes
1answer
62 views

A set $A \subseteq \mathbb{R}$ is closed if and only if every convergent sequence in $\mathbb{R}$ completely contained in A has its limit in A

Real analysis is a topic I'm unfamiliar with and I'm confused on how to write proofs on them. In order to prove that: A set $A \subseteq \mathbb{R}$ is closed (1) $\iff$ Every convergent sequence in ...
0
votes
2answers
49 views

Prove that if g is injective, f is injective

$f \colon A \to \mathbb R$ be a function (where $A$ is some set) and define the function $g \colon A \to \mathbb R$ as $g(x) = 3 (f(x))^2 + 1.$ Prove if $g$ is injective then $f$ is injective How do ...
2
votes
3answers
105 views

Limit of $(1+ x/n)^n$ when $n$ tends to infinity [duplicate]

Does anyone know the exact proof of this limit result? $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$
1
vote
1answer
54 views

Help prove $f:X \rightarrow Y$ is an injection $\Leftrightarrow$ $f:X\rightarrow Y$ is a surjection when $|X|=|Y|$

I need to prove: Given non-empty finite sets $X$ and $Y$ with $|X| = |Y|,$ a function $X\rightarrow Y$ is an injection if and only if it is a surjection. The hint given is to use the pigeonhole ...
4
votes
3answers
42 views

Does continuity of $f$ imply $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$?

I'm struggling to prove or disprove that the continuity of $f$ implies $f^{-1}(\bar A)\subset\overline{f^{-1}(A)}$. $f:X\to Y$ is a map between metric spaces $(X,d),(Y,d')$ while $\bar M$ denotes the ...
0
votes
1answer
45 views

Help to prove the existance of a function

Let $f:X \rightarrow Y$ be a function. Prove that there exists a function $g:Y \rightarrow X$ such that $f \circ g = I_Y$ if and only if $f$ is a surjection. I need help on proving the following: ...
0
votes
1answer
35 views

Proving theorems on relations

I came across the following three statements about relations. I understand why the statements are true, but I am not sure how to demonstrate them mathematically. In the following, $A,B$ are sets and ...
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 $w_1$. ...
1
vote
1answer
108 views

Question about proving that a finite intersection of big unions is a big union of finite intersections

Let $I_{1}$,...$I_{k}$ be index sets and for each $1 \leq m \leq k$ and each $j \in I_{m}$, let $U_{j}$ be a set. Prove that: $$(\bigcup\limits_{j_{1}\in I_{1}}U_{j_{1}}) \cap ... ...
0
votes
2answers
40 views

Proving that any common multiplication of two numbers is a multiplication of their least common multiplication

Im trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multipe is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple ...
1
vote
1answer
49 views

Strange proposition in probability book for conditional probability

I found the following proposition (15.1) in the probability book of Heinz Bauer: Let us given that $X$ is a numeric random variable on $(\Omega,\mathcal{A},P)$ which is non-negative / ...
1
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1answer
110 views

Proving the properties of big union of unions for indexed sets

Let $I$ be an index set, and for each $i \in I$, let $J_{i}$, be another index set. For each $i \in I$ and $j \in J_{i}$, let $U_{j}$ be a set. Set X = $\bigcup\limits_{i\in I}J_{i}$. Prove that: ...
1
vote
1answer
40 views

Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
0
votes
3answers
41 views

How do I prove $x^n < x^m$ when $m > n$ and $x > 1$

Title I made an attempt at it here: $x^n < x^m$ when $m > n$ and $x > 1$, $m$ and $n$ are naturals so divide both sides by $x^n$ so $1 < x^{m-n}$ but here i am stuck. Please help!
0
votes
2answers
46 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
0
votes
1answer
49 views

Question about proofs with topological spaces

How would I write a proof for this example? Let $(X, \tau_{1})$, $(Y, \tau_{2})$ and $(Z, \tau_{3})$ be topological spaces. A function ${f}: X \rightarrow Y$ is said to be continuous if for every V ...
0
votes
1answer
48 views

Question about writing proofs for limit

I intuitively understand proof with limits, but I'm not sure on how to write a formal proof for this example. For each $n \in \mathbb{N}$, let $a_n$, $b_n$ be real numbers. Also, let $a_{\infty}$, ...
0
votes
1answer
73 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
1answer
35 views

Question about written proof for geometric summation

Suppose $\alpha$ $\ne$ $\beta$ $\in \{0, 2\}^\mathbb{N}$ Prove that $$\sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \ne \sum\limits_{k = 0}^\infty\frac{\beta(k)}{3^k}. $$ This is the written proof ...
0
votes
1answer
55 views

Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
0
votes
1answer
44 views

How to prove that a given map is an injection?

Let $g:\mathbb{N_{m_1-1}}\rightarrow \mathbb{N}_{m_1}$, where: $$g(i) = \left\{ \begin{align} i & \text {, for } i<i_0 \\ i+1 & \text{, for } i \ge i_0 \end{align}\right.$$ and $i_0 ...
0
votes
5answers
53 views

Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

What I've got so far is this: Base case: n = 3 then $3 *2 * 1 = 6$ and $3^3 = 27$ $\therefore 6 < 27, 3! < 3^3$ So the base case is true. So if we assume $n! < n^n$ (n > 2) $(n + 1)! = ...
1
vote
2answers
31 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: ...
1
vote
4answers
46 views

Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

I'm trying to prove this by induction but I'm running into some trouble. The base case is $0$, so, $a^0 = 1$, the inequality holds true Being new to induction, I don't exactly know what to do for ...
-1
votes
2answers
57 views
2
votes
1answer
50 views

Question about proof with geometric sums

I am confused on how to write proofs for geometric sums. I think that using the well ordering principle to find the least n $\in$ $\mathbb{N}$ with $\alpha(n)$ $\ne$ $\beta(n)$ would be a good ...