All Questions
7
questions
0
votes
0
answers
32
views
Partitioning $\mathbb{N}$ into infinitely many infinite pairwise disjoint sets [duplicate]
Find an infinite collection of infinite sets $A_1,A_2,A_3\ldots$ such that $A_i\cap A_j=\emptyset$ for $i≠j$ and $$\bigcup_{i=1}^{\infty}A_i=\mathbb{N}.$$
My Attempt:
Let $p_i$ be the $i^{\textrm{th}}...
- 2,145
1
vote
1
answer
75
views
How do you prove that there exists a highest element of any finite, nonempty subset of Natural Numbers? Is the following algorithmic proof valid?
Since the given set, $C \subset \mathbb{N}$ is non empty, hence by well ordering principle there exists $\alpha \in C$ which is the lowest element in C. Also, since the set $C$ is finite, $\quad \...
- 194
0
votes
2
answers
78
views
Is this an adequate proof that any non-empty subset of N has a minimal element?
I am trying to improve my own standards for proof writing, but I cannot attend school, so I do not have the luxury of being able to speak to professors or peers to verify my attempts. In the proof ...
- 11
2
votes
1
answer
220
views
Determine whether or not the following structure $(P,S,1)$ is a Peano System
First this is how the book define as a Peano System.
By a Peano System we mean a set $P$, a particular element $1$ in $P$, and a singulary operation $S$ on $P$ such that the following axioms are ...
1
vote
0
answers
91
views
Show that: The set of all finite subsets of $\mathbb{N}$ is a countable set
Show that:
The set of all finite subsets of $\mathbb{N}$ is a countable set
My attempt:
Lets define the set $M:=\lbrace K : K \subset \mathbb{N} \wedge |K|<\infty \rbrace$
We now show that $|M|=|...
- 1,011
2
votes
1
answer
254
views
Set theory: $n$ is a set in the naturals, if $x$ is in $n$, is $x$ also a natural number?
Im having trouble with a homework question from my Set Theory class. The question is, Let $n$ be a set and an element of the natural numbers. If $x$ is an element of $n$, is $x$ also an element of the ...
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