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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2
votes
1answer
31 views

Help formalizing this proof about a continuous, one-one function.

I'm having a bit of trouble getting the language on this proof right, though I think I have the idea correct. I have the function $f\colon D \rightarrow {\bf R}$ where $D = [a,b]$. The function is ...
0
votes
1answer
41 views

Prove that there is a 1-1 correspondence between the set of subgroups of $\mathbb{Z}/N \mathbb{Z}$ and the set of the positive divisors of $N$

Im interested in the above Proof, is because I have the intiuition that it is not true at all, because for example, all the primes have exactly 2 positive divisors 1 an themselves, How Can I prove or ...
0
votes
4answers
38 views

Proof without using induction that a number is divisible by 6

Prove without using induction that all numbers of the form $6|8^n - 2^n$. I need a brush up on subtracting numbers with the same base but different exponent. So far I have $8^n - 2^n = 2^{3n} - ...
1
vote
3answers
72 views

If $f$ is continuous, $f(1) >1$ and $f(x+y)=f(x)f(y)$, then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
0
votes
1answer
41 views

Well defined Functions on Congruence classes

Could someone please confirm my logic or point me in the right direction? Thank you. 1) Is the function $f : [\mathbb{Z}]_p \to [\mathbb{Z}]_p$ given by $f([n]_p) = [n^2]_p$ well defined? 2) Is the ...
1
vote
2answers
34 views

Show $f(rx) = [f(x)]^r$ where $r\in\mathbb{Q}$.

Consider the function $f$ with the following properties: $$(1) \lim_{x\rightarrow 0} f(x) =1,$$ $$(2) f(x+y)=f(x)f(y),$$ $$ -\infty<x,y<\infty.$$ Show that $f(rx)=[f(x)]^r$ where ...
1
vote
2answers
52 views

Proving directly that ($a+b)^3 \equiv a^3 + b^3 \mod 3$

Assuming a and b are integers, I must prove directly that: $$ (a + b)^3 \equiv (a^3 + b^3) \mod 3 $$ First, my peers and I made the mistake of assuming what we are trying to prove and thus failed. ...
0
votes
1answer
46 views

Limit proof for rational function $\frac{1}{x}$

A while ago I posted another one like this with a incorrect approach, please see this one! Is this an accurate proof for limits for the function $\frac{1}{x}$ $\displaystyle \lim_{x\to1} \frac{1}{x} ...
0
votes
1answer
46 views

Proof of Functions!

Question: Let $ f\colon Z \to Z $ and $ g\colon Z \to Z $ be two functions. Prove that the following are functions. a. $h(x)\colon Z \to Z $ defined as $h(x) = f(g(x))$ when $g(x)$ is an onto ...
0
votes
1answer
48 views

Equivalence classes of multiples of 3

I'm having a little trouble wrapping my head around the elements of the equivalence classes using the following definition: for m, n in N, define m ~ n if m^2 - n^2 is a multiple of 3. 1) List four ...
3
votes
1answer
28 views

Eulerian circuit with no isolated vertex is connected

This is my first question (ever), and I am pretty new to math. So I ask for patience and understanding in advance. So this is the proof I came up with: Consider $G = (V,E). $ By definition of ...
1
vote
2answers
33 views

Well-defined functions on equivalence classes

Could someone please explain how to develop a proof and the reasoning behind the following: My professor said that to determine if a function is well-defined, we check to see if equivalent elements ...
3
votes
2answers
84 views

How to derive an proof for this infinite square root equation?

Here is continuous square root, namely: $\sqrt {1 + a \sqrt {1+b \sqrt {1+c\sqrt {1 +...}}}}$= any integer Find $a,b,c,d,e,f,...$ in general Uh, very interesting algebra pre-calculus problem, yet ...
0
votes
1answer
29 views

Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff $(a_n)_{n=m'}^{\infty}$ does.

Is my proof correct? Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Let $c$ be real number. and let $m' \geq m$ be an integer. Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
0
votes
0answers
28 views

Show that both $*$ and $.$ operation are same.

Let $G$ be a topological group with identity $x_0$. Let $\pi_1(G,x_0)$ is a fundamental group with the usual $*$ operation. If we define $(f.g)(s)=f(s)g(s)$ $\forall s\in [0,1]$ $\forall f,g\in ...
1
vote
0answers
32 views

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \to \sqrt{x}$

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \rightarrow \sqrt{x}$ for $x > 0$. Let $\epsilon > 0$ be arbitrary, want to find $N \in \mathbb{N}$ such that $n \geq N \Rightarrow ...
0
votes
2answers
56 views

Proof by induction failure if assumption is wrong?

I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start ...
0
votes
1answer
29 views

If $x\in \mathbb{R}^n$ and is a unit vector, why is $\sum\limits_{j,k=1}^n |x_j||x_k| < n^2$?

This is an excerpt of a larger proof: Other pertinent information: $A$ is a positive definite $n \times n$ matrix The set $C$ is the unit sphere I don't get the last inequality: $\gamma \sum ...
3
votes
1answer
52 views

Show that if A is diagonalizable, then sin^2(A) + cos^2(A) = I. Does this identity also hold for nondiagonalizable matrices?

Show that if A is diagonalizable, then $\sin^2(A)+\cos^2(A)=I$. Does this identity also hold for nondiagonalizable matrices? This is what I got so far: $$ e^{iA}= \cos A +i\sin A \\ \cos A= ...
1
vote
1answer
63 views

Is this an accurate limit proof for sine?

$\displaystyle \lim_{x\to 0} \frac x{1 + \sin^2(x)} = 0$ proof $$\left|\frac x{1 + \sin^2(x)}\right| < \epsilon$$ $$|x| < \delta$$ Let's require $|x| < 1$ so therefore, $$\sin^2(|x|) + 1 ...
0
votes
1answer
45 views

Limit proof for $1/x$ (as $x \to 1$)

Prove $\lim_{x\to 1} \frac{1}{x} = 1$ Using $\epsilon-\delta$ $|\frac{1}{x} - 1| < \epsilon$ for some $|x - 1| < \delta$ $|\frac{1}{x} - 1| = \frac{|1-x|}{|x|}$ Lets require $|x - 1| < ...
0
votes
1answer
17 views

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$ I want to do the $n - 1 \rightarrow n$ induction step. But I'm confused as to what my base case is. Usually if I want ...
1
vote
3answers
28 views

Prove that a sequence has a limit

Sequence $a_{n}$ satisfies $|a_{n}| \leq n$ for all $n \in \mathbb{N}$. Let sequence $b_{n} = \frac{a_{n} + 5}{n^{2} + a_{n}}$, prove that $b_{n}$ has a limit, and find it. I know that $b_{n}$ has a ...
1
vote
1answer
53 views

How to interpret the logic of an “or” in a matrix proof.

I am trying to learn to better interpret the meaning of equations and that is the purpose of this question, not just to find the proof, but to find the logical flow of the proof and understand it. I ...
1
vote
2answers
27 views

Next step in proof of sets

Proposition to prove : (A-B)∩(B-A) = 0 So, I understand why this is 0, I'm just not sure what propositions should be used in proving so. I have this so far 1)(A-B)∩(B-A) :Premise ...
3
votes
1answer
25 views

If $H,K$ are subgroups of $G$, and $G$ is finite, prove that $[K\colon (H\cap K)]\leq [G\colon H]$

Let $H,K$ be subgroups of a finite group $G$. Prove that $[K\colon (H\cap K)]\leq [G\colon H]$. This is what I have: $[K\colon (H\cap K)] = |\left\{ a(H\cap K) \mid a\in K\right\}|$ $[G\colon H] = ...
0
votes
1answer
61 views

If a sequence ${a_n}$ is monotonically increasing. then $\lim_{n \to \infty} a_n = \sup{(a_n)}$

Can you please tell me if my proof is correct: If a sequence ${a_n}$ is monotonically increasing. Then $$\lim_{n \to \infty} a_n = \sup{(a_n)}$$ Proof: $$a_n\leq a_{n+1}\leq \sup(a_n)$$ Assume ...
0
votes
2answers
25 views

Proof by induction with variable other than $n$

1) Prove that $(1+x)^{n} \geq 1 + nx$ for every $n \in \mathbb{N}$ and $x \in (-1, \infty)$ Base case: Usually for the base case I just take $n = 1$ but since there's another variable $x$, I wasn't ...
2
votes
1answer
53 views

Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable

I have to prove the following Suppose $f$ is Riemann integrable on $[a,b]$ and $1/f$ is bounded on $[a,b]$. Prove that $1/f$ is Riemann integrable on $[a,b]$. My attempt: Since $1/f$ is bounded ...
2
votes
2answers
44 views

Show that $\int_{x=a}^{x=b} f'(x) g(x) dx=f(b)g(b)-f(a)g(a)-\int_{x=a}^{x=b} g'(x)f(x)\, dx$

I have to prove the following: Suppose $f$ and $g$ are differentiable on $[a,b]$ and $f'$ and $g'$ are integrable on $[a,b]$. Prove that $f'g$ and $g'f$ are integrable on $[a,b]$ and that of: $$ ...
1
vote
4answers
64 views

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$

Let $G$ be a group. Prove the equivalence relation: If $H$ is a subgroup of $G$, let $a \sim b$ iff $ab^{-1} \in H$ To prove an equivalence relation my guess is to show that reflexivity, ...
6
votes
2answers
46 views

Proving a complete and totally bounded metric space is compact.

I'm having trouble writing down the details of this proof formally. Statement: Suppose $(X, d)$ is a metric space that is complete, and totally bounded (i.e., for every $\epsilon > 0$, ...
1
vote
0answers
31 views

How to prove unicity in a disjunction of $n$ propositions

Let's suppose I have the propositions $\varphi_1, \varphi_2,...,\varphi_n$ and I want to prove that there happens exactly one of them. How do you do it? To do it the hard way I guess we first need to ...
1
vote
6answers
65 views

Prove that $gcd(a, b) = gcd(b, a-b)$

I can understand the concept that $\gcd(a, b) = \gcd(b, r)$, where $a = bq + r$, which is grounded from the fact that $\gcd(a, b) = \gcd(b, a-b)$, but I have no intuition for the latter.
0
votes
1answer
38 views

There exists a positive real number $u$ such that $u^3 = 3$

Modify the Theorem that states There exists a positive real number x such that $x^2 = 2$. Show that there exists a positive real number $u$ such that $u^3 = 3$. So far, I have come up ...
1
vote
2answers
77 views

Proving logic statements

For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋). Use this definition to prove or disprove the following with a structured proof technique: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋. ...
1
vote
2answers
41 views

Proving $\forall x (A\to B) \to(A \to \forall x B):x\notin \mbox{free}(A)$ in a Hilbert system where it is not an axiom

I have no idea whether this question is way too specific or whether something similar has already been asked (we still need to work out a way to search for formulas I guess). Anyways here I go: I ...
2
votes
0answers
13 views

Derivation of higher order bessel function in terms of lower order functions

I am really stuck trying to prove this.. ((x^-p)Jp(x))’ = -(x^-p)Jp+1(x) ---(1) Can someone please help how to actually prove this step by step, because whichever notes i see, they prove ...
0
votes
0answers
19 views

Justify each step in the following proof of Proposition 3.9 (b). I like to know if I'm in the right track and I'm also missing a few of them

Proposition 3.9(b): If a ray emanates from an interior point of triangle ABC, then it intersects one of its sides. proof (a) Let r be a ray emanating from an interior point D. (b) The ray AD ...
1
vote
2answers
22 views

Prove that $(1+x)^n ≥1+nx$ for all $x>-1$ and $n=1,2,\ldots$

Prove that for every real number $x > −1$ and every $n = 1,2,\ldots,$ $$(1+x)^n ≥1+nx.$$ I don't know where to begin so I haven't tried anything.
0
votes
1answer
46 views

Justify each step in the following proof of Proposition 3.9 (a). I like to know if I'm in the right track and I'm also missing a few of them

Proposition 3.9(a): If a ray r emanating from an exterior point of triangle ABC intersects side AB at a point between A and B, then r also intersects side AC or side BC. proof. (a) Let r= array XD ...
0
votes
1answer
15 views

How to prove a subset is over an interval using the definition of subset.

Determine whether {x ∈ R : x2 < 9} is a subset of the interval [0, 9]. Prove your answer using the definition of A is a subset of B.
7
votes
2answers
249 views

Proving a palindromic integer with an even number of digits is divisible by 11

I'm in an introductory course for discrete math so I'm a novice at English proofs. I'm not sure if my reasoning here is valid or if I'm using modular arithmetic correctly. Specifically the line I ...
0
votes
2answers
42 views

Prove that sequences $\frac{a_n}{b_n} = 0$

Let $(a_n)$ and $(b_n)$ be positive real sequences such that $\lim \limits_{n \to \infty} \dfrac{a_n}{b_n} = 0$ and $(b_n)$ is bounded. Prove that $\lim \limits_{n \to \infty} a_n=0$. Proof: ...
0
votes
1answer
27 views

Proof of a limit formula

If $h(x) = f(x)/g(x)$ $lim(x->b) f(x) = L$ $lim(x->b) g(x) = M$ Prove that $lim(x->b) h(x) = L/M$ Sorry for the terrible latex. ONLY FORMAL PROOFS! For every $\epsilon > 0$ Since ...
0
votes
1answer
62 views

Limit Delta-Epsilon proof

Prove $\lim_{x \to a} 2x = 2a$ Using the formal proof, not informal. So we know $2|x - a| < \epsilon$ We need to find some $\delta$ We only need to prove there IS SOME $\delta$ right? Only ...
0
votes
4answers
87 views

For every $x \in [\frac{\pi}{2},\pi]$, $\sin(x)+\cos(x)\geq 1$. Prove rigorously by contradiction.

For every $x \in [o,\frac{\pi}{2}]$, $\sin(x)+\cos(x)\geq 1$. How do you prove this rigorously by contradiction? I understand you start by assuming that $\sin(x)+ \cos(x)<1$ and prove this is a ...
0
votes
1answer
20 views

The number of ways to paint a red tile in a grid.

here's the question: "You have nine tiles arranged into a three by three square mosaic. If you color each tile red or blue with equal probability, what is the probability that there exists a two by ...
1
vote
2answers
34 views

Induction on the number of marbles in a heap.

Here is the problem in full: "A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
1
vote
1answer
54 views

Prove: $\{α_1,…,α_n\} ⊨ α$ iff $\{α_1,…,α_{n−1}\} ⊨ (α_n→α)$.

Recently began my second logic course and have been surprised at how very, very different it is from the first one. My main struggle is that we have to prove things all the time, and I've never learnt ...