Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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Prove that an elementary row operation on the product of two matrices .

In my math textbook there is a method for finding inverse of a given matrix using row operation but the book does not give any proof for that. I searched for the proof of the on the internet and had ...
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How may one solve problems over expressions like $(2+px)^6$ without the binomial theorem?

A friend of mine posed a problem on a mathematics discord server. The coefficient of the expansion of $(2+px)^6$ is $60$. Find the value of the positive constant $p$. I immediately thought of ...
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Find if the rational log of a rational number is rational?

I'm currently working on a project and need to find whether or not $\log_\frac{a}{b}\Bigl(\frac{c}{d}\Bigr)$ can be expressed as $\frac{f}{g}$. Is there a computationally efficient way to do this? ...
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Need help writing proof

I need help writing a proof for a question from Velleman's "How to Prove It". The question is as follows: Prove that for all real numbers x and y there is a real number z such that x + z = y - z ...
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Can different variables refer to the same object without an identity rule stated explicitly?

For example, $\forall x(Qx\rightarrow \exists y(Py\wedge Rxy))$, if the Universe of discourse only contained one object, can this sentence be true?
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quantifier negation proof with natural deduction?

How can I derive ∃𝑥¬𝑃(𝑥)⊢¬∀𝑥𝑃(𝑥)? I know that I need to derive some sort of contradiction, but what do I assume?
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Prove that $\exists !c \in \mathbb{R} \exists ! x \in \mathbb{R} (x^2 + 3x + c = 0)$

This is an exercise from Velleman's "How To Prove It". I am struggling with how to finish the final part of the uniqueness proof, so any hints would be appreciated! a. Prove that there is a ...
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1answer
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Learning Latex typesetting [closed]

I would like to learn how to use Latex, on this site and other forums like AoPS. What are the resources you would recommend for someone with no prior experience?
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Do we need to rigorously prove why a set is non-empty with non-negative integers?

For proofs using Well-Ordering Principle(WOP), can we prove the set is a non-empty set of nonnegative integers simply by "stating"? Eg of what I mean by "stating": Given any integer $n$ and any ...
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1answer
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Proof Explanation: If $m \in n$, $\exists p \in \omega$ for which $m + p^+ = n$

Synopsis In Exercise 4.23 of Enderton's Elements of Set Theory, we are asked to show that if $m \in n$, $\exists p \in \omega$ for which $m + p^+ = n$. This seems like an obvious statement, but I ...
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Proof Without contradiction $x^2 = 2$ has no rational solutions

I am in the process of learning real analysis and I was wondering if there was a way to prove no $x \in \mathbb{Q}$ satisfies $x^2 = 2$ without a proof by contradiction.
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Proving that the spectrum of $A\in End(V)$ is contained in the the set of roots of polynomials for which $p(A)=0$

I want to prove the lemma: $$\tag{1}\sigma(A)\subseteq R(p)\ \forall\ p:p(A)=0$$ Where $\sigma(A)$ is the spectrum of eigenvalues of $A\in End(V)$ and $R(p)$ the set of roots of the polynomial $p$....
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Let $f:[0,1] \rightarrow [0,1]$ continuous. Prove that $x\in [0,1]$ exists that $f(x) = x$ [duplicate]

MY TRY: So if the function is continuous it means: $$|x-a|<\delta$$ Because the function is bounded we already know that $\delta \in (x-1,x+1)$ or that $\delta \leq 1$ and $a \in [0,1]$ There ...
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Proof of Inequality : $\frac{x}{y+z} + \frac{y}{z+t}+\frac{z}{t+x}+\frac{t}{x+y} \ge 2$ [closed]

How can i prove it? Looks like Nesbitt's inequality but i cant. If $x,y,z,t \in \mathbb R$ and $x,y,z,t \ge 0$, then $$ \frac{x}{y+z} + \frac{y}{z+t}+\frac{z}{t+x}+\frac{t}{x+y} \ge 2 $$
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Show that the function $f(x)=\frac{x-1}{2(x+1)}$ is continuous in $a=3$

What I've done is the following. $$\biggl|\frac{x-1}{2(x+1)}-\frac{3-1}{2(3+1)} \biggr|<\epsilon$$ By some calculation, I got $$\biggl|\frac{3x-5}{2(x+1)} \biggr|<\epsilon.$$ This is greater ...
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42 views

Convexity of $f(x) = \sum\limits_{i=1}^n \sqrt{x_i}$

Suppose we have some $x = (x_1, x_2, .. x_n)$ such that $x_i \geq 0$ and $\sum x_i = 1$. Let $f(x) = \sum\limits_{i=1}^n \sqrt{x_i}$. Is $f$ a convex function of $x$? It appears to be the case from ...
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Signed measure: verification

Let $f\colon X\to[-\infty,+\infty]$ a measurable function such that $$\int_Xf^+\;d\mu<\infty\quad\text{or}\quad \int_Xf^-\;d\mu<\infty,$$ then $$\nu(E)=\int_Ef\;d\mu$$ is a signed measure. I ...
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Suppose $P(x)$ is a polynomial with real coefficients satisfying the condition $P(\cos \theta + \sin \theta) = P(\cos \theta − \sin \theta)$

Suppose $P(x)$ is a polynomial with real coefficients satisfying the condition $P(\cos \theta + \sin \theta) = P(\cos \theta − \sin \theta)$ for every real $\theta.$ Prove that $P(x)$ can be expressed ...
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Proving that the dot product $\mathbb{R}^n\times \mathbb{R}^n\mapsto\mathbb{R}$ is bilinear

My lecturer presented a proof of the fact that the dot product $\mathbb{R}^n\times \mathbb{R}^n\mapsto\mathbb{R}$ is bilinear. To show that the map is linear with $y$ fixed he wrote the following: $$\...
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Let $x,y>1$ be coprime integers and $g>0$ a real number such that $g^x,g^y$ are both integers. Is it true that $g\in\mathbb N$?

Let: $x, y\ $ be coprime integers greater than $1$ $g \in \mathbb{R}^+$ $g_,^x \ g^y \in \mathbb{N}$ Proposition: $g \in \mathbb{N}$ I have not managed to prove it. Via the fundamental theorem of ...
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Set Operation Implication Proof

If A\B is a subset of C, then A\C is a subset of B. I'm super confused and not sure how to do this. I attempted proof by contradiction but not understanding what to do. Sorry, this is a duplicate I ...
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Proof regarding inverse images and unions of sets

I'm currently working through Analysis by Tao and I just did an exercise but I'm not sure if it's correct. The question is Let $f: X \to Y$ be a function from one set X to another set $Y$, and ...
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Prove that if $x \in A$ and $x \not\in (B \cap C^c)$, then $(x \in A$ and $x \not\in B)$ or $(x \in A$ and $x \not\in C^c)$

This is part of a larger proof where I am attempting to show that $$A \setminus (B \setminus C) = (A \setminus B) \cup (A \setminus C^c).$$ I have already shown that $(A \setminus B) \cup (A \...
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1answer
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Isomorphism between $S_n$ and a subgroup of $S_{n+1}$

Let $S_n$ be the symmetric group on $n$ letters. Now, I wish to show that $S_n$ is isomorphic to the subgroup of all the elements of $S_{n+1}$ that leaves $n+1$ fixed. Let $\sigma \in S_n$. Define ...
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PROPOSED PROOF for: If $p$ is a prime number such that $p|ab$, then $p|a$ or $p|b$.

Since $p|ab$, then $\exists \alpha \in\ \mathbb{Z}$ such that: $$\alpha p=ab$$ By dividing both sides by $a$, we obtain: $$(\alpha \div b)p=a \Rightarrow\ p|a \ \ ( \text{Similarly to prove that} \ ...
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Let $a$ & $b$ be non-zero vectors such that $a · b = 0$. Use geometric description of scalar product to show that

Let $a$ & $b$ be non-zero vectors such that $a \cdot b = 0$. Use geometric description of scalar product to show that $a$ & $b$ are perpendicular vectors. What I've done so far is to state ...
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For every $x$ and $y$ there exists $z$ such that $x + y = z$ [closed]

How to verify the truth validity of this predicate in the domain of $\mathbb{N}$ and then with $\mathbb{Z}$? I know that it is true, because there always exists such $z$, but how to write it down ...
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Is this a valid proof of the euclidean algorithm?

I created a proof of the euclidean algorithm and since I'm not really familiar with the field of discrete mathematics I'm asking for a revision : Having two natural numbers $a$ and $b$ we follow this ...
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1answer
22 views

Existence of a sequence such that random variables converge almost surely

I face a problem doing the proof of the following statement: Let $X_1, X_2,...$ be random variables. Show that there exists a sequence $(c_n)_{n \in \mathbf{N}} \subset (0, \infty)$ sucht that $X_n ...
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Prove that $\alpha \cup \{\alpha \} = \text {inf}\ \{\beta\ \mid\ \beta \gt \alpha \}.$

Let $\alpha$ be an ordinal.Then prove that $$\alpha \cup \{\alpha \} = \text {inf}\ \left \{\beta\ \mid\ \beta \gt \alpha \right \}.$$ I know that if $\alpha$ is an ordinal then so is $\alpha \cup \{...
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Example of basis

Excuse me , can you see this question , the collection of all open intervals (a,b) together with the one-point sets {n} for all positive and negative integers n is a base for a topology on a real ...
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1answer
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Limit proof by contrapositive

I'm writing a proof for the statement : "If a function $f$ has a limit at $p$, then for every $\epsilon >0$ there is a $\delta>0$ such that $\vert f(x_1) - f(x_2)\vert < \epsilon$, whenever $...
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Question about numerical methods [closed]

How can i show that Δ(log(f(x)))=log(1+(Δf(x)/f(x)) ?
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Examples of basis

Excuse me , can you see this question, For each positive integer $n$ , let $S_n=\{n,n+1,\ldots\}$ . The collection of all subsets of natural number which contain some $S_n$ is a base for a topology on ...
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Prove $\sigma$-sublinearity of Lebesgue upper-integral

I'm having trouble understanding the proof of the $\sigma$-sublinearity of the Lebesgue upper-integral given in my notes. The property: Let $f,g:\mathbb{R}^d \to [0,+\infty[$ and $c\ge 0\,\,(c\in\...
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How can I prove that the determinant of a matrix formed of polynomials of degree n-2 or smaller is zero?

The following matrix is formed by polynomiais of degree $n-2$ or smaller and $a_1,\cdots,a_n$ belong to $\mathbb{R}$. How can I prove that it’s determinant is zero? I thought about using the fact that ...
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Understanding Fraleigh's proof of that the set $R[x]$ obeys associativity w.r.t. multiplication

I am trying to understand Fraleigh's proof of the fact that the set $R[x]$ of all polynomials in an indeterminate $x$ with coefficients in a ring $R$ obeys the associative law for multiplication. Here ...
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How to prove if $(A \cup B) - (A \cap B) = A$ is true, then $B = \emptyset$?

I have figured out that this is true, but I'm not quite able to prove it. I have tried direct proof, contrapositive and contradiction. For contradiction, I assumed that B is not the empty set. Then,...
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Advice on proving statements without if-then form?

I do not know if this is the right place to ask but Googling failed. Whenever I do a "prove something" question, I try to formally restate in an if-then form. But when a statement can't be naturally ...
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are there rules that cant be rigorously explained?

this is sort of a shower thought question that came to mind, and I would prefer if it wasn't taken violently seriously, but are their rules that cant be rigorously explained. like for example the ...
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1answer
26 views

Finding the Domain?

The set ${{x:\left|x+\frac{1}{x}\right|>6}}$ equals the set My Approach We know,IF $|f(x)| \geqslant a, \quad ;a>0$ then $f(x) \geqslant a \quad \cup \quad f(x) \leqslant-a$ $\frac{x^{2}+1}{...
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Let $A,B,X$ be sets such that $A\cup B = X$ and $A \cap B = ∅$. Show that $A = X\backslash B$ and $B = X\backslash A$.

I'm trying to prove this Let $A,B,X$ be sets such that $A\cup B = X$ and $A \cap B = ∅$. Show that: (1) $A = X\backslash B$ and (2) $B = X\backslash A$. My proof is Let $x \in A$. We ...
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Binary Relation [closed]

I am not sure how to solve these problems. Can someone guide me through the steps. Prove that for any binary relation $R$ and $S$, the following identities hold: (a) $(R^{-1})^{-1}=R$ , where $R^{-1}...
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How do I prove that $\forall x\in\mathbb{R}:(\forall y\in\mathbb{R}: y>0 \implies 0\leq x\lt y)\implies x=0$

When reading this proof of the uniqueness of limits of sequences I stumbled across this argument (in the last few lines): We have a $y\in\mathbb{R}$ with $y>0$. (Originally called $\epsilon$) We ...
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1answer
32 views

If $f(x)$ is continuous on $[0,1], \text{ and } 0\leq f(x)\leq1, \forall x \in [0,1], \text{ prove } \exists t \in [0,1] \text{ s.t. } f(t) = t$

My thinking is that $f(x)$ has to intersect with function $g(x) = x$ at some point, but I don't know how to prove this.
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Proof of the existence of a unique linear transformation

I want to prove the following lemma: Let $B$ be a basis for $V$ and let $T_B: B \rightarrow W$ be a map. Then there exists a unique linear map $T_V: V \rightarrow W$ which extends $f,$ that is, such ...
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0answers
26 views

General notation for continuous and discrete random variables

I'm trying to represent a set of variables that model different features for a statistic model. Let us say that each sample in my data domain as $m$ features that can be modeled with $X_1,...,X_m$. ...
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1answer
36 views

Show that the direct sum $f\oplus g:X\to\textbf{R}^{2}$ defined by $f\oplus g(x) = (f(x),g(x))$ is uniformly continuous.

Let $(X,d_{X})$ be a metric space, and let $f:X\to\textbf{R}$ and $g:X\to\textbf{R}$ be uniformly continuous functions. Show that the direct sum $f\oplus g:X\to\textbf{R}^{2}$ defined by $f\oplus g(x) ...
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1answer
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Smallest possible dimensions of a piece of paper after one fold.

So I've been thinking, can someone explain mathematically if I started with a square of paper dimensions 1x1, what the smallest single side dimension could be reached after one fold. My gut instinct ...

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