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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Real Number recursion

Let $T_{n}$ be a sequence of real numbers such that $T_{n+1}\ge T_{n}^{2}+\frac{1}{5}$ Prove that $\sqrt{T_{n+5}}\ge T_{n}$. I rearranged the first expression in terms of $T_{n}$, but seemingly to no ...
0 votes
0 answers
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Mixing saddle points of a zero-sum game

I am fairly new to proof writing so I would appreciate if you could help me out! Suppose that $(X^*, Y^*)$ and $(X^0, Y^0)$are saddle points of a zero-sum game with payoff matrix $A$. Show that $(X^*...
-2 votes
1 answer
44 views

infinite series proof for negative numbers

I know this is not to be done with induction. I simplified the RHS to $1-\frac{1}{1+n}$, there should be some terms that cancel, but I'm stuck at how to do this. Perhaps using the condition $n<1$?
-5 votes
1 answer
52 views

Sum of rational inequalities proof

I wasn't given anything to use before i encountered this problem, so i want to figure out how to make headway on this proof. I tried combining the denominators but that didn't get me anywhere.
2 votes
3 answers
1k views

Prove by contradiction that a circle chord is no longer than its diameter

Can anyone help me with this homework question of mine? I'm actually new to proofs. Here's the question, "Prove, by contradiction, that no chord of a circle is longer than a diameter." My ...
2 votes
2 answers
2k views

Foot of perpendicular proof.

I know there is few answer on Foot of perpendicular, but my doubt is different, so please don't mark this as the duplicate. My book says: Foot of the perpendicular from a point $(x_1,y_1)$ on the ...
-2 votes
0 answers
30 views

Chinese Remainder Theorem w/ Multiples

First, find all multiples of 9 that have remainder 6 when divided by 16, 17, and 18. Second, find all multiples of 6 with the remainder of 12 when divided by 16, 17, and 18. I assume we do something ...
0 votes
0 answers
30 views

Using a direct proof, how should I approach this question?

For integers $x$ and $y$, if $xy$ is even, then either $x$ or $y$ must be even. I know how to solve this in both a contradictive and contrapositive manner but can not quite figure out how to solve it ...
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1 answer
64 views

Logic behind contrapositive proofs that involves De Morgan's Laws

Suppose $a,b\in\mathbb{Z}$. If both $ab$ and $a+b$ are even, then both $a$ and $b$ are even Proof by contrapositive. Propositions: $P$: $ab$ is even $Q$: $a+b$ is even $R$: $a$ is even $S$: $b$ is ...
0 votes
3 answers
71 views

Question in validity of proof that $9|n^3 + (n+1)^3 + (n+2)^3$

I wanted to prove that 9 divides the cubes of 3 consecutive integers. I represented this as: For $n \in \mathbb{Z}, 9|n^3 + (n+1)^3 + (n+2)^3$ By the definition of divisibility, if $9|n^3 + (n+1)^3 + (...
0 votes
1 answer
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How do I go about proving if a|b then $a^n|b^n \forall n \in \mathbb{N}$

I'm trying to prove that if a|b then $a^n|b^n, \forall n \in \mathbb{N}$. I know $a|b \rightarrow b = aq, q \in \mathbb{Z}$ by the definition of divisibility. I can then (I think) exponentiate both ...
1 vote
2 answers
102 views

How to argue that two outcomes are equally likely?

When it comes to the concept of "equally likely," students are not told what makes two outcomes equally likely, but are instead shown what it means via examples. "When flipping a coin, ...
2 votes
2 answers
8k views

Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too [duplicate]

I'm trying to figure out how to prove that if $p$ and $p^2+2$ are prime numbers then $p^3+2$ is a prime number too. Can someone help me please?
19 votes
4 answers
3k views

proving that the area of a 2016 sided polygon is an even integer

Let $P$ be a $2016$ sided polygon with all its adjacent sides perpendicular to each other, i.e., all its internal angles are either $90$°or $270$°. If the lengths of its sides are odd integers, prove ...
0 votes
1 answer
2k views

Proof of the Crossbar theorem

A teacher asked me to prove the well known Crossbar theorem. I tried it in the following way:- Given: If $D$ is in the interior of $\triangle ABC$, then prove that $\overrightarrow{AD}$ intersects $\...
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1 answer
42 views

How to prove the implication $(\neg q \rightarrow (\neg p \lor \neg q))$?

I am working on a proof for the statement: Suppose $x, y \in \mathbb{R}$. If $5|x$ and $5|y$, then $5|xy$ Let's denote the propositions as follows: $P: 5|x$ $R: 5|y$ $Q: 5|xy$ Logically, this is ...
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0 answers
38 views

How to resolve nested if then statements resulting from using the conditional identity.

If I want to turn the "p or q" part of an "if p or q then r" statement into an if then statement using the conditional identity then how do I resolve the nested if then statement ...
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0 answers
290 views

Proof verification: Every convex function on $(a,b)$ is continuous .

A real-valued function $f$ defined in $(a, b)$ is said to be convex if $f( tx+ (1 - t)y) \leq tf(x) + (1 - t)f(y)$ whenever $a < x < b, a < y < b, 0 < t < 1$ I want to Prove that ...
5 votes
1 answer
135 views

Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers?

Can this proof that $\sqrt{2}$ is irrational be rewritten using only integers? Most proofs that $\sqrt{2}$ is irrational start with assuming that $2=\dfrac{a^2}{b^2}$ and derive a contradiction. For a ...
14 votes
10 answers
3k views

How does one perform induction on integers in both directions?

On a recent assignment, I had a question where I had to prove a certain statement to be true for all $n\in\mathbb{Z}$. The format of my proof looked like this: Statement is true when $n=0$ "...
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1 answer
36 views

Proving the general formula for conditional expectation of a Poisson Process

I am studying a course on Stochastic Processes and encountered the following proof exercise on Poisson Processes: If $N$ is a Poisson Process with intensity $\lambda$, then for $0<s<t$ where $k ...
4 votes
5 answers
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Prove that $\sqrt{8}$ is irrational in different method

I tried to prove that $\sqrt{8}$ is irrational. I said let $\sqrt{8}$ be rational then $\sqrt{8}$ = $a/b$ where $a$ and $b$ are relatively prime. Then $2\sqrt{2}=a/b$ , and $\sqrt{2} =a/(2b)$. it is ...
-2 votes
2 answers
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Prove If $a \equiv b\pmod{n}$ and $c\equiv d\pmod{n}$, then $a + c \equiv b + d \pmod{n}$ [closed]

Using the definition of congruence $\pmod{n}$, prove that if $a \equiv b\pmod{n}$ and $c \equiv d \pmod{n}$, then $a + c \equiv b + d \pmod{n}$. I've seen a couple of different examples of how to do ...
0 votes
0 answers
43 views

Given two linear transformations T1 and T2, show that range T1 = range T2 if and only if there is an invertible operator S such that T1=T2S.

I am working through Axler's Linear Algebra Done Right section 3D problem 5 which states "Suppose $V$ is finite-dimensional and $T_1, T_2 \in \mathcal{L}(V,W)$. Prove that range $T_1=$ range $T_2$...
0 votes
1 answer
97 views

Is $(\phi\implies\psi) \iff ((\phi\land\psi)\lor \neg\phi) $ provable in the following hilbert calculus without contraposition or reductio?

Is $(\phi\implies \psi) \iff ((\phi\land \psi)\lor \neg \phi) $ in the following hilbert calculus without contraposition or reductio ad absurdum? If so, how would I go about proving it, and if not, ...
0 votes
0 answers
32 views

Is my proof using induction correct. $(\forall n \in \mathbb{N}, n\geq 5) 2^{n+3} < (n+1)!$

I'd appreciate if somebody could check whether my proof (mathematical induction) is correct. $$(\forall n \in \mathbb{N}, n\geq 5) 2^{n+3} < (n+1)!$$ (i) let $n=5$ $2^{5+3} = 2^{8} < (5+1)!=6!$ (...
1 vote
2 answers
1k views

Proving set statements

Let A = {x ∈ Z: x = 5a + x for some integer a} B = {y ∈ Z: y = 10b - 3 for some integer b} C = {z ∈ Z: z = 10c + 7 for some integer c} Prove or disprove the ...
2 votes
4 answers
762 views

Proving $ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}$.

Give an algebraic and a combinatorial proof for the following identity: $$ \sum _{k=0} ^m \binom nk \binom{n-k}{m-k} = 2^m \binom {n}{m}.$$ For the combinatorial argument, use the analogy ...
1 vote
0 answers
21 views

$P (A_1)= P (A_1\cap A_2)+ P(A_1\cap A_2^c)$ (Proof verification)

Is my proof well writen and correctly stated? Let $(\Omega,\mathscr Z ,P)$ be a space of probability and $A_1,\dots,\in \mathscr Z$, then: $P (A_1)=P (A_1\cap A_2)+P (A_1\cap A_2^c)$ Dem: By ...
0 votes
3 answers
79 views

Proof that if A and B are connected sets, $A \cup B$ is connected iff $(\bar A \cap B) \cup (A \cap \bar B) \neq \emptyset$

The problem says: Let $(X,\tau)$ be a topological space so that $A,B \subset X$ are connected sets in said space. For $M\subset X, \bar M$ refers to the closure of $M$. Prove that $A \cup B$ is ...
0 votes
0 answers
55 views

A simple algorithm to find the inverse [duplicate]

Someone showed me such an algorithm. I'm going to explain his method with an example. Let's say we're looking for $$37^{-1}\in \mathbb Z/63\mathbb Z$$ First, we make the successive divisions of Euclid'...
1 vote
1 answer
1k views

Proof: center of a tree lies on the longest path

How can I make a proof of this property? I mean, given a weighted tree (with positive costs), how can I proof that the center of such a tree lies on the longest path? I read to the first answer of ...
0 votes
2 answers
65 views

Show that the sequence $(1+N^2)/[N(1+N)]-5/6$ is positive and increasing

I have the function \begin{equation*} f(N)= \frac{1+N^2}{N(1+N)}-\frac{5}{6}, \end{equation*} where $N\in\mathbb{Z}_{++}/\{1, 2\}$. I want to show that $f(N)>0$ and that $f(N+1)>f(N)$ for ...
0 votes
1 answer
49 views

Struggling with proving an elementary atomic equation. Please critique my proof.

I am learning how to prove and disprove atomic equations. I have put together a proof where I know how I want to prove the theorem but can't quite get it to flow correctly. Please critique my syntax, ...
5 votes
2 answers
3k views

how to prove $G$ is an abelian group under $*$ (called the real numbers mod 1)

Let $G = \{x \in \mathbb{R}~|~0\leq x < 1\}$ and for $x,y \in G$ let $x*y$ be the fractional part of $x+y$ i.e $x*y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$. ...
0 votes
2 answers
113 views

Is there a visual proof for why this property of a matrix is true?

Let's say we have three equations written in their standard form: \begin{align} a_1x + b_1y + c_1 = 0 && (l_1) \\ a_2x + b_2y + c_2 = 0 && (l_2) \\ a_3x + b_3y + c_3 = 0 && (...
4 votes
5 answers
598 views

Trigonometric Identities: Given that $2\cos(3a)=\cos(a)$ find $\cos(2a)$

Given that $2\cos(3a)=\cos(a)$ find $\cos(2a)$. $2\cos(3a)=\cos(a)$ I converted $\cos(2a)$ into $\cos^2(a)-\sin^2(a)$ Then I tried plugging in. I know this is not right, but I have no clue how to ...
2 votes
3 answers
69 views

Proof that $\cos(x) = \cos\left(\frac{\pi}{2}\right)\implies x = \frac{\pi}{2} + k\pi$

I know that $$\cos(x) = \cos\left(\frac{\pi}{2}\right)$$ yields to $$(1)\qquad \qquad \qquad x = \frac{\pi}2 + 2k\pi$$ or $$(2)\qquad \qquad \qquad x = -\frac{\pi}2 + 2k\pi$$ which (as I found out in ...
28 votes
0 answers
2k views

Is there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?

Theorem: The group $(\mathbf Z/(p))^\times$ is cyclic for any prime $p$. Most proofs make use of the fact that for $r\geq 1$, there are at most $r$ solutions to the equation $x^r=1$ in $\mathbf Z/(p)$...
2 votes
1 answer
1k views

Proving the relation between the Dirichlet eta function and the Riemann zeta function [closed]

The problem I am trying to solve is: I need to prove the relation between the Dirichlet eta function and the Riemann zeta function $\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$. But I have no clue ...
0 votes
1 answer
31 views

Need hints (advice) to prove $(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$

I'm trying to prove this ( source : my uni's textbook says that it's trivial). $$(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$$ So far, I've managed to get ...
0 votes
0 answers
74 views

Difficulty with proving or refuting a claim related to a recursive sequence

I am trying to prove a certain claim about a recursive sequence. The sequence is defined as follows: $$ S_{n}^{k}(a) = \begin{cases} \sqrt{a} & \text{if } n = 1, \quad a \in \mathbb{Z}^{+}, \, a ...
3 votes
2 answers
445 views

What is wrong with this proof by contradiction?

My professor tells me that a proof by contradiction structure below is poor: Proof. To prove: A implies B. So, assume A (in order to prove that B follows). Assume not B (in order to arrive at a ...
0 votes
0 answers
59 views

Prove that $(\forall a,b,c \in \mathbb{R}, b \neq 0 ) c\frac{a}{b}=\frac{a}{b}c=\frac{ca}{b}$ [closed]

Prove that $(\forall a,b,c \in \mathbb{R}, b \neq 0 ) c\frac{a}{b}=\frac{a}{b}c=\frac{ca}{b}$ The first equality can be solved using commutativity of multiplication, right? So: $c\frac{a}{b}=\frac{a}{...
5 votes
4 answers
15k views

Show that the closure of $A$ is the intersection of all closed sets containing $A$, topology proof needed

I want to show that given $(X, \mathcal{T})$, we define $\overline A = \{x \in X| \forall U \in \mathcal{T}, x \in U \implies U \cap A \neq \varnothing\}$ (definition of closure from Munkres), then ...
0 votes
1 answer
117 views

Theorem 7, Section 4.5 of Hungerford’s Algebra

If $R$ is a ring with identity and $A_R$, ${}_{R}B$ are unitary $R$-modules, then there are $R$-module isomorphisms $$A\otimes_R R\cong R \text{ and } R\otimes_R B\cong B$$ Sketch of proof: Since $R$ ...
13 votes
7 answers
5k views

Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.

I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$." This is my attempt of proving it:- Assume that $x=p/q$...
4 votes
6 answers
27k views

prove there is no smallest positive rational number

How would I prove there is no smallest positive rational number? what is the best method to prove this statement?
0 votes
0 answers
42 views

When can a sequence be substituted for its limit

I'm new to proof-writing in Real Analysis, and came across the following problem: Prove $\lim(a_nb_n)=\lim(a)\lim(b)$. Let $\lim(a), \lim(b)$ exist and $\lim(a)=L_a, \lim(b)=L_b$ The first thing I was ...
15 votes
7 answers
4k views

Prove $1+2\sqrt3$ is not a rational number

How would I go about proving $1+2\sqrt 3$ is not a rational number assuming $\sqrt 3$ is not a rational? Would direct proof be the easiest? Total beginner here, any insight would be appreciated.

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