Is $\mathbb Q = \{\text{set of all numbers periodic from some point on}\}$ ? I am pretty sure that $\{\text{set of all numbers periodic from some point on}\} \subset \mathbb Q$ but i don't know how to prove. I didn't find any proof about $\mathbb Q \subset \{\text{set of all numbers periodic from some point on}\}$. Maybe it is wrong. Thanks for the help !
10186
I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly appreciated.
56314
Q: Is every element of $\mathbb Z/p\mathbb Z$ represented by a prime number? More generally, let $m, n \in \mathbb Z$ be coprime. Is there a prime number congruent to $m$ modulo $n$?   the affirmative answer is given by .     A: Yes, in fact there are infinitely many such primes.   I don't want to prove Dirichlet's theorem. Is it obvious that there always exists at least one such prime?
109899
$\mathbb R$ is an extension of $\mathbb Q$.  Why the elements of $\mathbb Q(2^{1/3})$ have the form $a+b\sqrt[3]2$?  More specifically why $\mathbb Q(2^{1/3})=\{a+b\sqrt[3]2+c\sqrt[3]4:a,b,c \in\mathbb Q\}$?
63087
Suppose $A,B,C$ are sets.  Prove that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$.  I proved that for $x\in A\setminus B$, $x\in A\cup C$ and for $x\in B\setminus A$, $x\in B\cup C$. How do I show that $A\cup C=B\cup C$, and that $A\mathbin\triangle B\subseteq C \iff A\cup C=B\cup C$?
31074
I want to prove that every open interval has the same cardinality of $\Bbb R$.  The question is:  Is it enough to prove that any open interval is uncountable? If I prove it, can I say that this interval has the same cardinality of reals?  I know that I can get a bijection using the tangent function, but I am not allowed to use trigonometric functions.  I've proved that |$(a,b)$|=|$(0,1)$| so I may prove that |$(0,1)$|=|$\Bbb R$| but I did not find the bijection without using trigonometric functions.
61055
I would like to prove the existence of $b \in \mathbb Q$ such that $a&lt;b^2&lt;c$ for any given $a,c \in \mathbb Q$ with $a,c&gt;0$  I want to use the statement above to prove a statement in a  I thought that '$b$' must be exist. But, in my opinion, $\sqrt{}$ can't not be used because $\sqrt{a}$ or $\sqrt{c}$  may not exist in $\mathbb Q$ for some $a$ and $c$.  I couldn't find a clue to prove the statement before the real number is constructed from $\mathbb Q$. Would you help me to prove that?  Thanks all for replying and pointing out errors.
50392
I am very new to proofs so please excuse any trivial errors.  In lecture we were told that:     A set $\mathbb S$ is called finite if there exists a one-to-one    mapping (bijective mapping) between $\mathbb S$ and the set $\mathbb  N_n=\{0,1,2,3,...,n-1\}$ for some natural number $n$. A set is called    countable infinite if there exists a bijective mapping between    $\mathbb S$ and the set $\mathbb N=\{0,1,2,...\} $ of all natural    numbers   I want to answer the following questions:  1. Show that the set $\mathbb S=\{a,b,c\}$ is countable  Does countable mean countable infinite? An attempt from another post on this site: For all triples $(a,b,c)$ there exists an according sum $S=a+b+c \implies ...$ But wouldn't the sum be a surjection between $\mathbb S$ and $\mathbb N$ and not a bijection?  2. Show that the set of integer numbers $\mathbb Z=\{0,\pm1,\pm 2,...\}$ is countable infinite  Before I continue with this proof I have a question about mapping. The definition says that "if there exists a bijective mapping...". In this context, can a function $f(x)$ be considered a mapping? If yes, then can't I just define myself some map $$f:\mathbb Z\to \mathbb N$$  and let $\space f$ be: $$x \mapsto \lvert x \rvert$$  which would then automatically imply that $\mathbb Z$ is countable infinite? (or wouldn't it)?  3. Show that $\mathbb S=\mathbb N \times \mathbb N=\{x |x=(p,q),p,q \in \mathbb N\}$ is countable infinite.   I am not sure about this one but if my "intuition" from 2) turns out to be correct I can maybe define some sort of norm and show that it is a bijection between $\mathbb S$ and $\mathbb N$
50394
Denote by $\mathbb{N}$ the set of natural numbers and by $2^{\mathbb N}$ the set of all subsets of $\mathbb N$. Let $E$ be some subset of $2^{\mathbb N}$ such that for  every pair of elements in $E$, one is a subset of the other. Clearly, $E$ could be countable by considering $E=\big\{\{1,\cdots, n\}, n\in\mathbb N\big\}$. My question is the following: could we find an example that $E$ is uncountable? Many thanks for the answer!
101981
Here I am under the impression that $2\mathbb Z$ and $3\mathbb Z$ are the sets of even numbers and multiples of $3$ respectively and the operations are usual addition and multiplication. This is an exercise in Fraleigh Abstract Algebra.  I believe that $2\mathbb Z$ and $3\mathbb Z$ are isomorphic as groups on addition.  I started a proof by supposing that there was a ring isomorphism, $f$,  in hopes of finding a contradiction with $f(x\cdot y) = f(x)\cdot f(y)$.
88932
I am really confused from the definition.  How do we know that $\mathbb Q$ is not a free $\mathbb Z$-module?  In class people use it as a trivial fact, but I don't seem to understand.
25400
I need to find the cardinality of the set $S$ of all infinite subsets of $\mathbb{Q}$.   It's easy to prove $\operatorname{card} S=\mathfrak c$ if you first prove that the cardinality of the set of all finite subsets of $\mathbb{Q}$ is $\aleph_0$.   But is there some other proof for the statement without using the second one (about finite subsets)? Probably there is =)
66608
I have no idea how to prove it by using only what the book has talked about so far. Can anyone help? The proof shouldn't be using set theory as set theory is only mentioned in the following chapter. The proof should only make use of the addition of natural numbers, order properties of natural numbers, the trichotomy of order for natural numbers and principle of induction.  Proposition 2.2.14 (Strong principle of induction). Let $m_0$ be a natural number, and let $P(m)$ be a property pertaining to an arbitrary natural number $m$. Suppose that for each $m ≥ m_0$, we have the following implication: if $P(m')$ is true for all natural numbers $m_0 ≤ m' &lt; m$, then $P(m)$ is also true. (In particular, this means that $P(m_0)$ is true, since in this case the hypothesis is vacuous.) Then we can conclude that $P(m)$ is true for all natural numbers $m ≥ m_0$. Exercise 2.2.5. Prove Proposition 2.2.14. (Hint: define $Q(n)$ to be the property that $P(m)$ is true for all $m_0 ≤ m &lt; n$; note that $Q(n)$ is vacuously true when $n &lt; m_0$.)  Thanks!
19806
How to show that the sequence $$x_n, n \geq 0, x_{n+2}=|x_{n+1}-x_{n-1}|, n \geq 1$$ with $x_0, x_1, x_2$ positive integers, not all null,  is periodic?  I tried to pick up the square but obtained equalities not helped.  Thanks so much for any suggestion.
21587
I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it equivalent to find a bijection from $\mathbb{Q}$ to $\mathbb{Z}$?
120110
My question is, does there exist a set of subsets of the naturals, $S$, such that $S$ is uncountable, where for any two subsets $A,B \in S$ we have either $A \subset B $ or $B \subset A $?  Initially my thoughts were that we could construct a chain $A_{1} \subset A_{2} \subset \cdots $ and find a bijection by picking out one element which is 'new' to each set by axiom of choice and so we have bijected to a subset of naturals, however I'm not convinced this fully works.  Furthermore, I was thinking there may be a nice elegant solution by looking for such a set in either $ \mathbb{Q}^2 $ or $ \mathbb{Z}^2 $ for example, but any attempt seems to fail due to the 'ordering' part of the question.
123953
Can someone please show me how to prove this statement? $$\forall x\in\mathbb N (x &gt; 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$$ I can only assume $m = 3$ and when $x &gt; 1$.
53243
Is it true that every infinite finitely generated group contains an element of infinite period?  I think it is true, but I can't prove it.