For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

learn more… | top users | synonyms

7
votes
2answers
318 views

Questions on Proofs - Equivalent Conditions of Normal Subgroup - Fraleigh p. 141 Theorem 14.13

(1.) Why did Fraleigh shirk the proof for $(2) \implies (1)$? By dint of Arthur's comment, $(2) \iff \color{crimson}{gHg^{-1} \subseteq H} \quad \wedge \quad gHg^{-1} \supseteq H \implies ...
3
votes
1answer
130 views

Tricks, Multiply across Subset - Left Coset Multiplication iff Normal - Fraleigh p. 138 Theorem 14.4

Left coset multiplication is well defined by $(aH)(bH) = (ab)H \iff H \triangle G$. Given $H\leq G$, we wish to define a group structure on $G/H$ under suitable conditions. The natural way to do ...
0
votes
1answer
32 views

Need help proving an equality

I'm looking for help proving this equality: $\forall m:m \times 0 = 0 = 0 \times m$ Any help would be greatly appreciated. Thank you very much.
1
vote
2answers
93 views

Proving a summation involving binomial coefficients.

I need to prove the following inductively: (http://upload.wikimedia.org/math/9/e/5/9e57871ba17c1ad48e01beb7e1bb3bb9.png) $$\sum_{i=1}^{n} i{n \choose i} = n2^{n-1}$$ And for the life of me I can't ...
2
votes
1answer
89 views

Initial Segment Order Isomorphic to the Ordinal Numbers

Prove that every well-ordered proper class has an initial segment order isomorphic to the ordinal numbers, ON. I have a plan to prove this but it uses a recursive definition and induction which I do ...
1
vote
4answers
103 views

Why is $\langle \mathbb{Z}_4, + \rangle$ not isomorphic to $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$?

I'm having some trouble here, specifically with the idea of $\langle \mathbb{Z}_2 \times \mathbb{Z}_2, + \rangle$ as a group. Can anyone help me out with some explanations? Moreover, I generally ...
4
votes
1answer
300 views

Problems with fake proofs of limit of sequences

I can hardly imagine an easier example of the fact that my understanding of the topic is more than rusty. I will divide the question in two parts to make the reading easier: 1) Background; 2) ...
1
vote
1answer
56 views

Question about equlaity of two language, simple but tricky.

I found the following question tricky: If $A$ is a language, when will $A^*=A^+$? By definition, $$A^* = \bigcup^{\infty}_{i=0}A^i = A^0 \cup A^1 \cup A^2 \cup \cdots$$ $$A^+ = ...
1
vote
2answers
94 views

In triangle $ABC$ prove that $AB = 2BC$

In solving this proof I am NOT permitted to use any numerically related givens (i.e., the sum of all angles in a triangle is 180 degrees or in a right triangle side Asquared + side Bsquared = side ...
1
vote
1answer
30 views

Proof of Exponential function $a^n$

How can I prove that $$\sum_{n=N}^M a^n = \frac{a^M - a^{N+1}}{1-a}$$ for $a$ not equal to $1$ and $$\sum_{n=N}^M a^n = M-N+1$$ for $a = 1$
1
vote
2answers
24 views

A question about operations on languages.

I come across this problem on a book. It states that: for languages A and B, $(A\cup B)^* = (A^*B^*)^*$. I know that the definition of star closure is $\left(\bigcup^{\infty}_{i=1}\right)A^i$. But so ...
2
votes
5answers
389 views

proving zeros of a polynomial are not real

I'm working on a optimization problem and need to show that \begin{equation} \frac{1}{2}x^4 - x^3 -x + 100 = 0 \end{equation} has no real solution in order to prove certain properties about the ...
0
votes
1answer
29 views

When can the length of a line be equal to a circular function?

So, I'm having a bit of trouble trying to grasp this concept. I understand that a circular function like cosine is a ratio of two sides of a triangle in reference to an angle, however, one of my ...
2
votes
0answers
147 views

Stereographic projection from sphere to $\mathbb{R}^2$

This question is from my tutorial problem set: One way to define a system of coordinates for the sphere $S^2$ given by $x^2+y^2+(z-1)^2=1$ is to consider the stereographic projection $\pi:S^2-\{N\} ...
1
vote
1answer
109 views

What are the typical approaches to showing that some function sequence does not converge uniformly?

The following problem is from Munkres's Topology (Exercise 6 of Section 21 "The Metric Topology (continued)", 2nd edition). Exercise: Define $f_n : [0,1] \to \mathbb{R}$ by the equation $f_n(x) = ...
0
votes
3answers
218 views

Using rules of inference (Leibniz) to prove theorems.

Leibniz: If $A \equiv B$ is a theorem, then so is $C[p:= A] \equiv C[p:= B]$. Note: p is "fresh" means p doesn't occur in $A, B, C$. I am trying to understand how to use Leibniz rule of inference for ...
0
votes
1answer
54 views

Critical Points and Gradients/Derivatives

Plot the function $f(x)= 3+\cos(3x)-0.5\sin(5x)+0.2\cos\left(10x-\left(\frac{\pi}{4}\right)\right)$. Estimate how many critical points are on the interval $[0,2\pi]$. Consider $\mathbb{R}^{20} \to ...
1
vote
4answers
105 views

Prove or Disprove the existence of a basis

I'm asked to prove or disprove the existence of a basis $(p_0,p_1,p_2,p_3)$ of $F(t)(3)$ (Polynomials of degree at most 3) such that each of the polynomials $p_0,p_1,p_2,p_3$ satisfies the equation ...
4
votes
0answers
76 views

Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
0
votes
1answer
45 views

(Geometry) Proof type questions

Can someone please explain to me the given question and proof? otherwise I might just have to end up dropping my maths course because unfortunately I'm not understanding anything from my teacher. ...
0
votes
1answer
41 views

Regular language restricted to smaller alphabet is regular

Let $L$ be a regular language on some alphabet $\Sigma$, and let $\Sigma_1 \subset \Sigma$ be a smaller alphabet. Consider $L_1$ the subset of $L$ whose elements are made up only of symbols from ...
0
votes
3answers
87 views

Define $a\ast b=a+b+5$, and show $(\Bbb Z,\ast)$ is a group.

Let the set $\mathbb Z$ have the operation $*$ defined by $a * b = a + b + 5$ for all $a,b \in\mathbb Z$. Show this is a group. I understand how to prove closure and associativity. For ...
0
votes
1answer
51 views

Union and Intersection of families of initial segments

I'm trying to show that unions and intersections of families of initial segments are initial segments. An initial segment of a partially ordered set X is a subset of A such that, for every x$\in$X ...
0
votes
2answers
60 views

$|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$

Let $a,b,c ∈ \Bbb R$ Show that $|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$ Not sure where to start
3
votes
1answer
157 views

Prove! that $a+(1/a) ≥ 2$ and $a+(1/a) ≤2$

Let $a \in R$ If $a>0$, then $a+\frac1a\geq2$ If $a<0$, then $a+\frac1a\leq2$ This is how someone explained the first one to me but still not really sure about it. Proof: ...
0
votes
2answers
63 views

Prove that $a^2 \equiv b^2 \pmod q$ if and only if $a\equiv\pm b \pmod q$ for any prime numbers $q$.

Prove that $a^2 \equiv b^2 \pmod q$ if and only if $a\equiv\pm b \pmod q$ for any prime numbers $q$ homework question, please help.
0
votes
2answers
134 views

Gaussian integers - If $N(t)$ is an ordinary prime, prove then $t$ is a Gaussian prime

$\mathbb{Z}[i] = \{a+bi | a,b\in\mathbb{Z}\}$ Show that if $N(t)$ is an ordinary prime in $\mathbb{Z}$ then $t$ is a Gaussian prime in $\mathbb{Z}[i]$ (we say that $t\in\mathbb{Z}[i]$ is a Gaussian ...
0
votes
0answers
18 views

Prove that the following function of binary random variables is monotonic

Consider a binary random variable $y$ over the space $\mathcal{Y} = \{+1, -1\}$ such that $\Pr(y = 1) = q$. Consider also $r$ binary random variables $y^1, \ldots, y^1$ over the space $\mathcal{Y}$ ...
2
votes
3answers
114 views

Help with the algebra in for this number theory proof?

For all $n\geq 1$, prove with mathematical induction $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ So far.. I have substituted 1 and saw that the statement is ...
3
votes
2answers
48 views

Strong Induction - Understanding the lateral conditions.

I dont want the proof of this statement unless it is necessary for my questions, I just want some clarification. If cr = 1 would cr-1 = 0? How is cj 1 or 0? I understand cj is an arbitrarily ...
0
votes
1answer
60 views

Probability Theory proof question

Problem: In football, a coin known to be unfair is tossed to see who receives the first kickoff. Your team has a peculiar curse in that the probability of winning the game given that they won ...
1
vote
1answer
1k views

Prove that between every two rational numbers a/b and c/d that there is a rational number and there are an infinite number of rational numbers [duplicate]

So the full problem is Prove that between every two rational numbers $a/b$ and $c/d$ that: There is a rational number There are an infinite number of rational numbers I am having ...
5
votes
1answer
78 views

Question about sets of well-orders and isomorphisms between well-orders

I was thinking about this problem in trying to better get a feel for and understand well-orders (I'm trying to learn some set theory) - right now, I'm not really hoping for anybody to supply me with a ...
0
votes
4answers
620 views

Help me to Prove that log2 3 is irrational. [closed]

seemingly simple homework assignment, help? Was never the best with logarithms, how would I go about proving? Sorry the question read IRrational!
1
vote
1answer
29 views

Show that f is a symmetric relation on A

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here: Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ ...
0
votes
2answers
171 views

Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
1
vote
3answers
70 views

How can you prove this equality?

I am trying to figure out the following equality, but cannot seem to get anywhere. I tried integrating by parts, but that blew up when I set u = (log x)^n and tried to take log (0). I also tried ...
0
votes
2answers
66 views

Proof of $\lim_{n\to\infty}\left(\frac{a_n+b_n}{2}\right)^n=\sqrt{ab}$

Let $a_n$ and $b_n$ two strictly positive sequences such that $$\lim_{n\to\infty}a_n^n=a>0\qquad \lim_{n\to\infty}b_n^n=b>0.$$ I need to prove that ...
2
votes
4answers
90 views

Proof that $1+4^n+7^n$ is divisible by 3 for all $n \in \mathbb{N}$

So, as the title said I have to proof that $1+4^n+7^n$ is divisible by 3 for all $n \in \mathbb{N}$. I have to do it with induction. So I got my start, for $n=0$: We have $1+4^0+7^0 = 1+1+1 = 3$ and ...
1
vote
1answer
57 views

How to prove: If $a \to -\infty $ and $b$ is bounded from below by a constant $k\in\Bbb R^{>0}$, then the $a\cdot b\to -\infty$

I must proof the following, with $a: \Bbb{N} \to \Bbb{R}$ and $b: \Bbb{N} \to \Bbb{R}$ If $a \to -\infty\ (n\to\infty)$ and $b$ is bounded from below by a constant $k\in\mathbb R^{>0}$, then the ...
1
vote
3answers
90 views

Suppose x and y are coprime integers and z is a natural number. Prove that If xy is a zth power then x and y are both zth powers. [duplicate]

I'm supposed to use a prime factorization somewhere, and that the fundamental theorem of arithmetic is to be applied as well.
1
vote
1answer
89 views

Prove $(1 + \sqrt n)^{2/n} \leq (1 + 1/\sqrt n )^2$ for all natural $n$.

Prove $(1 + \sqrt n)^{2/n} \leq (1 + 1/\sqrt n )^2$ for all natural $n$ I may or may not have to use Bernoulli's Inequality in this question. I tried using Bernoulli's inequality on both sides of ...
1
vote
1answer
78 views

Proof concerning Latin squares

I'm asked to solve this problem : Let $R$ be an $r\times c$ partial Latin rectangle using the numbers $[n]= \{1,2,...,n\}$. Suppose that $r < n$ and $c < n$, and let $N(i)$ be the number of ...
8
votes
3answers
737 views

Proving the area of a square and the required axioms

I recently realized the area formula of all polygons, and most basic figures can be proven from the areas of square and rectangle. For example if we know the area of rectangle, we can the area formula ...
0
votes
4answers
66 views

Let x and y be integers, let x and y be greater than 0. Prove that the gcd (x/gcd(x,y) , y/gcd(x,y) = 1

Very confusing, not really sure how I'm supposed to deduce what $\gcd (x,y)$ is and how $$\gcd \left(\frac{x}{\gcd(x,y)} , \frac{y}{\gcd(x,y)}\right)$$ can be $1$?
0
votes
3answers
79 views

Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$

Problem Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$. Please help with the above. I have no idea what's going on. An explanation would be nice.
3
votes
2answers
362 views

Fallacious Induction Proof that All Integers are Equal [duplicate]

We're currently learning about induction in my real analysis course. I came up with the following proof that is obviously false but cannot quite figure out why it fails. Here it is... "Any set $S$ of ...
2
votes
1answer
147 views

Injection function and product of two exponential elements - homomorphisms -

[Fraleigh, p.133, ex. 13.7] Let $f_i: G_i \rightarrow G_1 \times G_2 \times \dots \times G_r$ be given by $f_i(g_i) = (e_1, e_2, ..., g_i, ..., e_r),$ where $g_i \in G_i$ and $e_j$ is the ...
0
votes
2answers
36 views

Groups Math Proof Help

Show that the indicated set $G$ with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. $G = \mathbb Z_5$ under addition mod $5$. I ...
4
votes
1answer
89 views

Image of Group Homomorphism is Finite and Divides |Domain of Group| - Fraleigh p. 135 13.44

Let $\phi: G \rightarrow G'$ be a homomorphism. Show that if $|G|$ is finite, then $|\phi[G]|$ is finite and divides $|G|$. Because $φ[G] = \{φ(g) \, | \, g ∈ G\}$, we see $|φ[G]| ≤ \quad |G|$ which ...