All Questions
Tagged with natural-numbers abstract-algebra
20
questions
2
votes
2
answers
143
views
For any $f:\mathbb{Z_+}\to \mathbb{Z_+}$ there exist $f(a)\le f(a+b)\le f(a+2\cdot b)$
I am preparing for the USAJMO, however I got stuck on the following question. I would really appreciate any help/hint.
For any function ( not necessarily having Bijectivity ) $f:\mathbb{Z^+} \to \...
- 129
0
votes
1
answer
33
views
Proving equivalence of two different induction definitions
enter image description hereI'm reading Stillwell's Elements of Algebra (see screen). I'm struggling with proving the equivalence of the versions of induction, specifically the step (II=>III).
If I ...
1
vote
1
answer
128
views
Identify natural numbers with integers
I'm studying Analysis from the book of Terence Tao and I have a problem with a paragraph where Tao establishes that we can set the natural number $n$ equal to integer $n—0$. He starts checking that $(...
- 71
0
votes
1
answer
81
views
Monoid actions - is this action inelegant?
I’m recently delving into abstract algebra, and I’ve attempted to devise a monoid action on the natural numbers. I think I must be missing something here—is there a better way to represent these same ...
- 335
1
vote
1
answer
72
views
Uniqueness of finite measures that agree on $\mathcal{H} \subset \mathbb{N}$
Defining
$$A_k = \{k, 2k, 3k, ...\}$$
for each $k \in \mathbb{N}$, and
$$\mathcal{H} = \emptyset \text{ } \cup \{A_k : k \in \mathbb{N} \}$$
I have been able to show that
$\mathcal{H}$ is closed ...
- 199
1
vote
1
answer
115
views
Constructing $(\Bbb N,+)$ via Peano function algebra duality.
In the next section we outline a $\text{ZF}$ construction of the natural numbers under addition using a 'duality' argument (the choice of the word duality is subjective and has no formal meaning).
...
- 11.5k
1
vote
0
answers
120
views
Prove that $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have the same cardinality iff m=n
In the following proof, $\mathbb{Z}_n= \{ 1,2,....n \} $
After $g$ is defined , they defined the composition $gof$ and then they defined $h$ by removing the last ordered pair $(n+1,m)$ from the ...
- 3,162
1
vote
1
answer
36
views
Well ordering of $\mathbb{N}$ with weak induction
In my Algebra course the well-ordering property of $\mathbb{N}$ has been proven just by standard ("weak") induction, but I can't understand the proof given by the lecturer.
The proof proceeds as ...
- 579
0
votes
1
answer
43
views
Proving Transitivity for the equivalence relation: $\langle a,b \rangle \sim \langle c,d \rangle \iff a+_{\mathbb{N}}d=b+_{\mathbb{N}}c$
For any $a,b,c,d \in \mathbb N$, I am trying to demonstrate that $\langle a,b \rangle \sim \langle c,d \rangle \iff a+_{\mathbb{N}}d=b+_{\mathbb{N}}c$.
I am trying to do this without invoking the ...
- 5,064
1
vote
1
answer
85
views
Possible monoidal structures on ordinal numbers
Suppose we take an initial segment $X$ of the class of ordinals such that $\sup X$ is a limit ordinal (for example $\mathbb{N}$). I know there are several ways of making $X$ into a monoid, like ...
- 4,099
0
votes
1
answer
113
views
Does a strictly increasing function satisfying this exist? [closed]
Is there a function $f: \mathbb{N}\rightarrow\mathbb{N}$ (strictly increasing) , such that the following is always true:
$$f(f(f(n))) = n + 2f(n)?$$
This is a homework problem from a contest math ...
- 353
1
vote
1
answer
38
views
Examples of commutative semirings satisfying $kb = hb + r \ \; \text{ iff } \; k = h \text{ and } r = 0$.
Let $\Bbb M$ be a commutative semiring. Setting $b = 1 + 1$, $\, \Bbb M$ also satisfies
P-1: For every $k,h \in \Bbb M$ and $r \in \{0,1\}$
$$\tag 1 kb = hb + r \ \; \text{ iff } \; k = h \text{ and ...
- 11.5k
3
votes
0
answers
119
views
Defining Addition of Natural Numbers as the Algebra of 'Push-Along' Functions
Let $N$ be a set containing an element $1$ and $\sigma: N \to N$ an injective function satisfying the following two properties:
$\tag 1 1 \notin \sigma(N)$
$\tag 2 (\forall M \subset N) \;\text{If } ...
- 11.5k
0
votes
1
answer
62
views
If $x,y \in \mathbb{N}$ then $x+y=0 \iff x=y=0$
Let $x,y \in \mathbb{N}$. The operation $(+)$ is defined by:
$$x+0=x$$ $$ x+(y+1)=(x+y)+1$$
Then prove that $x+y=0 \iff x=y=0$.
The second implication $x=y=0 \implies x+y=0 $ is simple and ...
- 1,349
-1
votes
2
answers
176
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How to show that $(S\cup\{0\},\ge)$ is order-isomorphic to $(S,\ge)$? [closed]
Let $S$ denote the set containing all the natural numbers that are not divisible by $ 2 $.
And define the binary relation $ \ge $ on two natural number $ m , n $ , $m \ge n $ if $m = k n $ , for ...
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