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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15
votes
3answers
537 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
4
votes
3answers
191 views

Do you need to find the domain of a function to prove injectivity?

I teach math and have had this debate with a fellow teacher. I believe the domain of a function is really not important to determine if a function is or is not injective if there is no "real life" ...
4
votes
3answers
190 views

Proving the Obvious

Is it normal that I have the hardest time when I'm trying to prove statements that are blatantly obvious on a visual and/or intuitive level? For instance, how does one go about formally proving the ...
1
vote
3answers
30 views

Wrting equations for work rate problems

Consider the following An experienced bricklayer can work twice as fast as an apprentice bricklayer. After the bricklayers work together on a job for 6 h, the experienced bricklayer quits. The ...
1
vote
0answers
42 views

Proof by induction for all positive integers

$\sum_{j=1}^n j^2 = \frac{n(n+1)(2n + 1)}{6}$ So I did the obvious and plugged one to show that $1^2 = \frac{1(1+1)(2(1) + 1)}{6}$ Now I am trying to show that $1^2 + 2^2 + ... + n^2 + (n + 1)^2 = ...
0
votes
0answers
23 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
0
votes
1answer
36 views

Need help with a proof:

What is the smallest element of {${3m+8n : m,n \in \mathbb{Z}, m \ge n \ge -4}$} Any help would be greatly appreciated. Thank you!
0
votes
0answers
32 views

Linear algebra proof help

Let $A$ and $B$ be similar matrices. Prove that the geometric multiplicities of the eigenvalues of $A$ and $B$ are the same. [Hint: show that, if $B=P^{-1}AP$, then every eigenvector of $B$ is of the ...
2
votes
1answer
71 views

Prove for every two sets $A$ and $B$

Prove for every two sets $A$ and $B$ that $A-B$, ${B-A}$ and $A \cap B$ are pairwise disjoint. I'm really stuck on this one. I know pairwise disjoint means no two elements in $A$ and $B$ are ...
0
votes
0answers
26 views

Proof sum of permutation

I'm trying to prove: $$P(N) = \sum permutation(A,N)=1 \tag{1}$$ for the particular choice of the set $A = \{ \mu_1, \dots, \mu_n, 1-\mu_1, \dots, 1-\mu_n \}$, where $i = 1, \dots, N$ . So for ...
0
votes
2answers
54 views

Prove 6 Divides n(n+1)(n+2) [closed]

Let n be an integer such that n >= 1. Prove that 6 divides n(n + 1)(n + 2). Not sure where to start, been stuck for a while
1
vote
0answers
31 views

Integral Domain and PID Proof

Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$. By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ ...
2
votes
1answer
33 views

Divisibility and Principal Ideal Domain Proof

Let R be a ring. Show that a|b iff $b \in (a)$ iff $(b) \subseteq $ (a). I first just want to write out what I know about this statement: a|b means that a divides b or a is divisible by b and there ...
0
votes
3answers
58 views

Maximal Proper Ideal is a field proof

Show that a proper ideal M of a commutative ring R is maximal if and only if R/M is a field. What I know: Because M is a proper ideal $M \neq R$. The ideal M is maximal if it is a maximal element ...
5
votes
0answers
202 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
2
votes
4answers
82 views

Proper Ideal and Units Proof

Show that an ideal of I R is proper if and only if it does not contain 1 iff and only if it does not contain any units. (1 is the identity element) I'll need to show 3 parts: (1) $\implies$ (2): ...
0
votes
0answers
31 views

Ideal Test Proof

Let $\emptyset \subset I \subseteq R$. Prove that I is an ideal of R if and only if $a-b, ra, ar$, $\in$ $I$ for all $a, b \in I$ and $r \in R$. I know that if I is an ideal in a ring R and $a \in ...
2
votes
1answer
51 views

Integral Domain and no nonzero divisors Proof

Prove that a commutative ring is an integral domain if and only if it has no nonzero zero divisors. I think my main problem is that I'm getting jumbled in the wording! By 'no nonzero zero divisors' ...
1
vote
1answer
27 views

Need help understanding a specific equality in this proof

Question. Let $f:\mathbb R\to \mathbb R$ be a uniformly continuous function. Show that there exists $a,b>0$ such that $|f(x)|\le a|x|+b,$ $\forall x\in\mathbb R$. Proof. Since $f$ is uniformly ...
0
votes
2answers
50 views

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M Is this a monotone sequence? Do I need to use Cantor's principle
0
votes
3answers
39 views

Proving the sandwich theorem for $\lim_{n \to \infty} c_n$ if $a_n \leq c_n \leq b_n$ and $a_n, b_n \to c$

Suppose $\lim\limits_{n \rightarrow \infty} a_n =\lim\limits_{n \rightarrow \infty} b_n = c$ and $a_n \le c_n \le b_n$ for all $n$. Prove that $\lim\limits_{n \rightarrow \infty} c_n = c$. How ...
0
votes
4answers
60 views

Proof roots of unity being in $\mathbb R$

Let $n \in \mathbb N$ even, and be $w,z \in \mathbb G_n$ primitives. Proof that $(w+z)^{n/2} \in \mathbb R$. Ok, as I didn't really know how to start, I tried several things, such using the Binomial ...
2
votes
1answer
35 views

Does all non-monotonic continues functions have $x_0 \in \mathbb{R}$ such that $f'(x_0)=0$?

Given $f\colon\mathbb{R} \to \mathbb{R}$, $f$ is differentiable on $\mathbb{R}$ and the $\lim_{x \to \infty}f(x)$ does not exists . show/prove formally that there exists $x_0 \in \mathbb{R}$ such ...
1
vote
1answer
37 views

Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$

$f(x)$ is continuous on $[a,b]$. Now we define a new function $M(t)$, for every $t\in[a,b]$ $$M(t) = \sup_{a \leq x \leq t} f(x).$$ Prove formally that $M(t)$ is continuous on $[a,b]$. (sup = ...
0
votes
1answer
58 views

Prove this limit

If $\lim_{x\to a}f(x)=L>0.$ Prove $\lim_{x\to a}\sqrt(f(x))=\sqrt(L)$. I know that we have: |$\sqrt(f(x)-\sqrt(L)|=|(f(x)-L)/\sqrt(f(x)+\sqrt(L)|\le|(f(x)-L)/L|<|(f(x)-L)|<\epsilon$. ...
1
vote
2answers
56 views

Linear surjective isometry then unitary

Basically what I'm trying to show is $\forall h_1, \ h_2 \in \mathscr{H}$ and $U: \mathscr{H} \rightarrow \mathscr{K}$ then $\langle Uh_1, \ Uh_2\rangle_\mathscr{K} = \langle h_1, \ h_2 ...
0
votes
5answers
150 views

Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$

Prove that $$\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$$ The exercise provides the following hint: $\,\,\displaystyle{n \choose k}={n\choose n-k}$. Any help?
0
votes
2answers
118 views

How to prove the equation |xy|=|x||y| if we assume x and y are real numbers by using analysis. [closed]

Prove that if x and y are real numbers, then |xy|=|x||y|. Hint check all the cases. I tried assuming the left hand side equals the right hand side if we remove absolute values. Also, tried using the ...
1
vote
1answer
25 views

Need help with a proof

Let $m, n \in \mathbb{N}$. If $n$ is divisible by $m$, then $m \le n$. So far I have: Let $m,n \in \mathbb{N}$ and assume that $n$ is divisible by $m$. Therefore, there exists $j \in \mathbb{Z}$ ...
0
votes
1answer
71 views

Writing my first mathematical paper

I'm an undergrad student at a community college, and I am currently taking differential equations. My professor is offering people to hand in math papers for either extra credit or even complete test ...
3
votes
3answers
129 views

Proof of something that doesn't exist

Let $\lfloor x \rfloor$ be the greatest integer function. Show that the $\lim_{x\to 2} \frac{1}{\lfloor x \rfloor}$ does not exist. So far I have: Assume the limit exists. Choose $\epsilon ...
0
votes
1answer
45 views

How to show that if $ p \equiv 1,3 \pmod 8$ then there exists a $u,v \in \mathbb Z: u^2 + 2v^2 = p$

I'm trying to show this statement: $$p \equiv 1,3 \pmod 8, \; \; \exists \; u,v \in \mathbb Z : u^2 + 2 v^2 = p.$$ I believed I proved it the other direction using the ring $\mathbb Z{\sqrt{-2}}$. ...
3
votes
2answers
65 views

Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function. Prove formally that $P$ is onto $\mathbb{R}$

Given $P\colon\mathbb{R} \to \mathbb{R}$ , $P$ is injective (one to one) polynomial function i need to formally prove that $P$ is onto $\mathbb{R}$ my strategy so far ....... polynomial function is ...
4
votes
2answers
109 views

formal proof from calulus

Given $f:R \to R$, $f$ is differentiable on $R$ and $\lim_{x \to \infty}(f(x)-f(-x))=0$. I need to show that there is $x_0 \in R$ such that $f'(x_0)=0$ I am trying to prove it by contradiction .... ...
0
votes
0answers
46 views

How to write down this proof about a graded ideal in multilinear algebra?

I have a very simple question, but since this is the first time I'm dealing with graded ideals and so on it seems more difficult than it really is. Suppose $V$ is a finite dimensional vector space ...
2
votes
2answers
47 views

My problem in understanding the minimal counterexample technique

If minimal counterexample method of proof is to assume to opposite of an argument is true and then finding a counterexample for the opposite and then concluding the validity of the original argument, ...
-2
votes
2answers
24 views

Proof Involving sum

Can someone point me in the right direction, should be able to figure it out. Let $x\in\mathbb{R}^+$. Prove that $x + \frac{1}{4x} \geq 1$ Thanks
-3
votes
2answers
133 views

Proof Involving Rational Numbers [duplicate]

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
1
vote
1answer
57 views

Linear Independent proof

In my Linear Algebra class we define Linear dependence as follows: If $F$ is a field and $V$ is a vector space over the field $F$. The set $A = {\lbrace v_1,v_2,...,v_k \rbrace}$ where ...
-2
votes
1answer
222 views

Rational Number Proof [duplicate]

Stuck on a tutorial question trying to study for a test. The question is : Consider the following statement: "Between any two different rational numbers, there are at least two different rational ...
1
vote
3answers
66 views

How is the Logarithm derived from the exponential function? (aren't they inverses?)

I've been learning logs in school, and my teacher, friend, and I are stumped on something. How does one derive the logarithmic function from the exponential function? My friend thinks Tayler Series ...
0
votes
1answer
35 views

Need help proving the following:

Any help at all would be great. Thank you very much. For all $m,n,p \in \mathbb{Z}$, If $p<0$ and $mp<np$ then $n<m$
20
votes
7answers
2k views

LaTeX/TeX Vs. Mathematica for Typesetting

I know Mathematica like the back of my hand, but I do not know a speck of $\LaTeX$ or $\TeX$. With regards to mathematical typesetting, is there something significant I can do in $\LaTeX$/$\TeX$ that ...
0
votes
3answers
128 views

Sum of $k {n \choose k}$ is $n2^{n-1}$

Proof that $\suṃ̣_{k=1}^{n}k {n \choose k}$ for $n \in \mathbb N$ is equal to $n2^{n-1}$. As a hint I got that $k {n \choose k} = n {n-1\choose k-1} $. I tried solving this by induction but, in the ...
0
votes
1answer
28 views

Group Order and Least Common Multiple

Let $G_1,G_2,...G_n$ be groups. Show that the order of an elements $(a_1,a_2,...a_n)$ $\in$ $G_1 \times G_2 \times ... \times G_n$ is lcm($o(a_1),...,o(a_n))$ I know I need to use the fact that the ...
1
vote
1answer
28 views

Multiplication cannot be obtained from zero, successor, and identity by composition without recursion

The task is to show that multiplication cannot be obtained by zero, successor, or identity functions by composition without using recursion at least twice. I'm primarily confused because it doesn't ...
1
vote
1answer
46 views

I don't understand part of a proof

I was reading a proof in my textbook today and couldn't figure out why this is true: $$ nq - mp = nq -mq +mq - mp$$ Any help would be appreciated.
0
votes
2answers
40 views

Need help with a math proof

Any help would be greatly appreciated. Let $m,n,p,q \in \mathbb{Z}$. If $0 < m < n$ and $0 < p < q$ then $mp < nq$.
1
vote
1answer
31 views

Proving Direct Sum

Claim. Let $V$ be a vector space over $F$, and suppose that $W_1$, $W_2$, and $W_3$ are subspaces of $V$ such that $W_1 + W_3 = W_2 + W_3$. Then $W_1 = W_2$. I know that this claim is false, but ...
0
votes
2answers
34 views

Proving convergence to a certain limit

Suppose that the sequence $(X_n)$ has the following property: there is a real number $a$ such that there are infinitely many $n$ for which $X_n = a$. Prove that, if $X_n$ converges at all, its limit ...