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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1answer
46 views

Proof on Open & Closed Sets

I just did a quick proof, and it seemed so simple that I wanted to check if it was correct. Prove that if you have a nonempty subset, $S$, of a domain $\Omega$, and $S$ is both open and closed, then ...
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2answers
91 views

Symmetric Positive Definite and Gradient Proof

I have the function $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x - \mathbf b^T \mathbf x$ where $Q$ is symmetric. I'm trying to show that solving $\nabla f(\mathbf x) = 0$ is equivalent to solving $Q ...
1
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2answers
61 views

Struggling with a proof that $-m=(-1)m$

Prove that For all integers $m$, $-m=(-1)m$. Any help would be greatly appreciated.
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2answers
580 views

Proving an infinite number of primes of the form 6n+1

The proofs given on other sites weren't that clear and used different methods that I have yet to learn. Prove that there are an infinite number of primes of the form 6n+1. The hint that was given ...
2
votes
1answer
27 views

Question with nonempty bounds and sets

Let $A$ and $B$ be nonempty sets of real numbers, bounded above and below. Prove that if $A\cap B$ is also nonempty, then $infB\leq supA$. So my train of thought goes like this: I'm picturing that ...
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2answers
49 views

Combinatorial Proof Question

I'm really iffy on combinatorial proofs in general and now that there is a sum, it's just confused me even more. Can someone try and walk me through this proof? $$ \binom{m + n}{r} = \sum_{k=0}^r ...
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0answers
87 views

Quadratic Symmetric Positive Definite Function Implies Convex

Consider the function $f$(x)=$\frac {1}{2}$*x*$^T$Q x. Show that if Q is symmetric positive definite, then $f$ is a convex function. Show that $\nabla f$(x) = Q x (provided Q is symmetric). I ...
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3answers
609 views

Proof that there are infinitely many primes congruent to 3 modulo 4

I'm having difficult proving this. As a hint the exercise to prove first, that if $a\lneqq \pm 1$ satisfies $a \equiv 3 (\textrm{mod}\ 4)$, then exist $p$ prime, $p \equiv 3 (\textrm{mod}\ 4)$ such ...
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0answers
38 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
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2answers
30 views

Discrete Structures proof

I'm in a discrete structures class and I'm having trouble with formulating ideas as to what I need to prove. Here's the question: Suppose you are trying to prove that, If a, b, and c are integers for ...
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4answers
123 views

$f:X\to Y$, $A,B,\subseteq X$. Show that $f(A\setminus B)=f(A)\setminus f(B)$ iff $f(A\setminus B)\cap f(B) =\emptyset$

I tried to prove this but I am not sure if its correct. Please help me out with any tips or advice on how to improve. Here it is: First let $f(A\setminus B)\cap f(B)=\emptyset$. Now $$f(A\setminus ...
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1answer
34 views

$h \circ g $ is 1-1 and onto $g$ is 1-1 and onto $h$ is onto - how to prove that $h$ is 1-1 too?

$h \circ g $ is 1-1 and onto $g$ is 1-1 and onto $h$ is onto I am trying to prove that $h$ is 1-1 too. $h \circ g(x1) $ =$h \circ g(x2) \rightarrow x1=x2 $ And due to the fact that $g$ is ...
0
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1answer
51 views

proof $B^{-1}MB$ is triangular

How to prove this? Theorem: Let $M$ be a matrix of complex numbers. There exists a non-singular matrix $B$ such that $B^{-1}MB$ is a triangular matrix. This is corollary from book Linear Algebra by ...
0
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1answer
83 views

Group Theory Exponent and Abelian Proof

Let G be a group such that $x, y \in G$ Show that, if $(xy)^2=x^2y^2$ or $(xy)^{-1}=x^{-1}y^{-1}$, then xy=yx. This can also be thought of as the exponent rule $(xy)^n$=$x^ny^n$ if xy=yx is true ...
2
votes
3answers
89 views

$n$-abelian Groups

Show that $(xy)^n=x^ny^n$ if $xy=yx$. I assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$. For the $n > 0$ case, can I use induction? For the base case I'll show that ...
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3answers
50 views

Formal expression for a proof

Yesterday I asked this question: Given: $f$ is Riemann integrable on $[a,b]$ and $f(x)\geq 0$ for all $x$. Prove that if \begin{equation} \int_a^b f(x) dx=0 \end{equation} and $f$ is ...
2
votes
6answers
120 views

Proof that if $a^n|b^n$ then $a|b$ [duplicate]

I can't get to get a good proof of this, any help? What I thought was: $$b^n = a^nk$$ then, by the Fundamental theorem of arithmetic, decompose $b$ such: $$b=p_1^{q_1}p_2^{q_2}...p_m^{q_m}$$ with ...
0
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1answer
47 views

Cardinality of a Monoid and Constant Functions

Let $X$ be a set. Show that $((M(X),\circ)$ has an absorbing element iff $|X|\leq 1$ iff $M(X)$ is commutative. In this problem $((M(X),\circ)$ is a monoid and M(X) is the set of all maps from X to ...
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1answer
72 views

proof of $\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial \frac{\partial f(x,y)}{\partial y}}{\partial x}$

I was at my physics class(electrodynamics).I saw a relation which frequently uses in my course.Relation is that $$\frac{\partial \frac{\partial f(x,y)}{\partial x}}{\partial y}=\frac{\partial ...
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2answers
68 views

Absorbing Element is a Unit

Show that an absorbing element of a monoid is a unit if and only if it is the only element. This is an if and only if proof so that means I have to prove it both ways: A implies B and B implies A. ...
0
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1answer
90 views

Proof: Lemma 5.6.2 - Elements of Real Analysis (C.G. Denlinger)

I reading the book "Elements of Real Analysis, C.G.Denlinger".. and I need the proof of Lemma 5.6.2: Thanks in advance!
0
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1answer
22 views

Prove as a direct theorem.

One of the question from my text book, it give a theorem and says that prove it as a direct theorem. For two statements A and B, the direct theorem is "if A is true, then B is true." In this case, is ...
2
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2answers
124 views

show that $2n\choose n$ is divisible by 2 [duplicate]

I tried using induction, but in the inductive step, I get: If $2n\choose n$ is divisible then I want to see that $2n +2\choose n +1$ $${2n +2\choose n +1} = (2n + 2)!/(n+1)!(n +1)! = {2n\choose n} ...
1
vote
1answer
68 views

Prove $\liminf(a_n) = \limsup(a_n) = \infty$

If $(a_n)$ is a sequence of real numbers, prove $(a_n)$ diverges to infinity iff $\liminf(a_n)=\limsup(a_n)= \infty$. I started with this but don't know if it is right or where to go next.... For ...
1
vote
1answer
28 views

Linear Algebra Proof with one-dimensional subspaces

Suppose that V is finite dimensional, with $dimV=n$. Prove that there exist one-dimensional subspaces $U_1,...,U_n$ of $V$ such that $$V = U_1 \oplus\dotsb\oplus U_n$$ My linear algebra is rusty, very ...
3
votes
3answers
96 views

Formal proof of: $x>y$ and $b>0$ implies $bx>by$?

Property: If $x,y,b \in \mathbb{R}$ and $x>y$ and $b>0$, then $bx>by$. What is a formal (low-level) proof of this result? Or is this property taken as axiomatic? The motivation for this ...
0
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1answer
36 views

Finite and Infinite Cardinality Representation

Let X be a set. Show that the cardinality of the set of finite sequences with elements from X has cardinlity $\aleph_o$ if X is finite and cardinality $|X|$ if X is infinite. I was given the hint ...
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2answers
102 views

prove or give a counter-example

I think I have solved it (please check) but I would like to see and (re)-learn how one writes a proper proof (including the mathematical signs) and little things (I might have missed), maybe even more ...
1
vote
1answer
55 views

Proving via axioms, that for given set $A$, $P(P(A))$ exists

The question itself: For a given set A, prove P(P(A)) exists. You may only use the axiom of pairing, axiom of union and axiom of empty set. This is how I solved it: Let A be the given set. ...
2
votes
2answers
95 views

Restate a logical claim using logical symbols

Proposition: Strictly between any two distinct rational numbers lies another rational number. How may I present this statement using logical symbols? My answer: $\forall x, y \in {\mathbb{Q}}. ...
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2answers
49 views

Contrapositive Proof: Specific Question! Need help!

I've been stuck on this question for a few days, please help me with this contra positive proof! Suppose that $x$ and $y$ satisfy $\frac 1 2 x + \frac 1 3 y = 1$. Prove that $x^2 + y^2 > ...
1
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1answer
53 views

Proof of variance of $S_n^2$

Let $$S_n^2= \frac{1}{n} \sum X_i - \overline{X}$$ I'm looking for a clear simple proof of the variance of this just for personal knowledge to derive the MSE later on any input would be appreciated.
0
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2answers
109 views

Cardinality of Integers, Positive Integers, and Rational Numbers all equal $\aleph_0$

Prove that $|\mathbb{Z}|=|\mathbb{Z}^+|=|\mathbb{Q}|=\aleph_0$ I am to use cardinal addition and multiplication to reduce this to finding an injection $\mathbb{Q}^+ \to ...
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3answers
52 views

Altering an Infinite Set does not change cardinality

Let X be an infinite set. Show that adding or subtracting a single point does not change its cardinality. I have a plan but need help writing the actual proof. I need to show that it doesn't matter ...
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0answers
114 views

The Toad and Frog Game - Proof by Inducation

Toads and Frogs is played on a 1 × n strip of squares. At any time, each square is either empty or occupied by a single toad or frog. Although the game may start at any configuration, it is customary ...
0
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1answer
52 views

Math proof involving open and closed intervals

So this is a multipart question, and I have a couple "theories" on how to attack it but still stuck in the infancy stages. Basically, not very far. For each real $r>0$ let $I_r$ denote the open ...
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2answers
52 views

Finding a real number c for polynomial (proof)

The question is to find a real number c for which $ x\ge c%+$ implies $$x^4-4x^3+7x-9 \ge1000$$. I was given the hint that $x>10$, then $4x^3<0.4x^4$, so $x^4-4x^3>0.6x^4$. Problem is, I'm ...
1
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1answer
49 views

Math Proof Question similar to reverse triangle inequality

Prove that for any real three numbers x,y,z, $$ |x-y||z| \le |y-z||x| + |z-x||y|$$ I am way overthinking this, there must be an easier solution to this. Any thoughts?
0
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1answer
56 views

Initial Segments and Isomorphism

Let $f:X \to Y$ be an order isomorphism and A be an initial segment of X. Show that $f[A]$ is an initial segment of Y. I believe this has something to do with Cantor's theorem but can't quite ...
1
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1answer
30 views

Showing that all regular languages are closed under reversal

I'm trying to show that $L^{reverse} = \{w^{reverse}:w \in L\}$ is a regular language. The first argument I can come up with is simply: if we have an NFA for $L$, then an NFA for $L^{reverse}$ can be ...
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3answers
128 views

Prove that $x^{−1}Hx $ is a subgroup of G [closed]

Let $H$ be a subgroup of a group $G$ and, for $x\in G$, let $x^{-1}ax$ denote the set $\{x^{−1}ax : a\in H\}$. Prove that $x^{−1}Hx$ is a subgroup of $G$.
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2answers
82 views

How to prove this is a partial order??

Let $R$ be the partial order on $\mathbb{N}$ (set of all natural integers) defined by: $$n \leq m \iff m = (2^k)\cdot n \;\text{ for some }k \in\mathbb{Z},\, k \geq0.$$ I know the basic idea on how ...
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0answers
73 views

Is this reasoning valid? (Probability)

It is often hard to tell if a line of reasoning is valid, especially when it leads to the correct answer. I'd like to hear your opinions on the following example from probability theory. I gave a ...
1
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3answers
53 views

Simple induction proof

Im having a lot of trouble proving by induction that $3^n + 5^n \geq 2^{n+2}.$ The base step is easy, but I don't seem to find the way to proof the inductive step.
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2answers
42 views

Prove that $a_n$ = $2^n$ + $(-1)^n$

I am given the sequence $a_0= 2$, $a_1= 1$ and $a_{n+2}= a_{n+1} + 2a_n$ How do I prove that $a_n$ = $2^n + (-1)^n$
2
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1answer
82 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 ...
3
votes
6answers
911 views

$(P\implies Q) \implies [(R ∨ P)\implies (R ∨ Q)]$ is a tautology

I'm currently trying to work on the proof for this tautology. But every time I derive the right side, I end up with a lone $R$ that will never cancel out. Like I always end up with $$(P\implies Q) ...
1
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2answers
89 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 ...
0
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1answer
135 views

Proving the fundamental period of tangent

I'm very new to math and proofs -- so I apologize if my math skills and vocabulary offends you. I have a question that states: Prove that PI is a fundamental period of the tangent function. I need ...
1
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1answer
41 views

On provability within minimal logic

In its most naive form my question boils down to this: when is a proposition that is provable "by contradiction" also provable "directly"? IOW, is it possible to know, a priori, that a ...