All Questions
Tagged with natural-numbers elementary-set-theory
152
questions
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204
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Can natural number be an ordered pair?
I’m supposed to prove that natural number $n$ cannot be an ordered pair. The definition for ordered pair is $(x,y) = \{\{x\},\{x,y\}\}$ and for the definition of natural numbers we use the definitions ...
- 1
2
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1
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254
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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 ...
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2
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Proof of the well-ordering principle
I tried to prove Well-Ordering Principle by myself, and I finally did it. However, I'm not sure if this proof is correct. Can anyone evaluate my proof?
Proof:
Since the set of natural numbers, $\...
- 123
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120
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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
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89
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Why is $\mathbb{N}$ well-ordered?
Define
$$0:= \emptyset$$
$$1:= \{\emptyset\} =\{0\}$$
$$2:= \{\emptyset, \{\emptyset\}\}=\{0,1\}$$
$$\vdots$$
$$n:= \{0,1, \dots, n-1\}$$
And put $\mathbb{N}:= \{0,1, \dots\}$.
Questions:
(1) ...
user745578
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80
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Is this proof using Cantor's Diagonal Argument correct?
We use $\sim$ to indicate to sets being bijective to eachother, i.e. having the same cardinality in this context. There exists $\psi:\mathbb{N}^2\mapsto\mathbb{Q}$ ,via $\left(a,b\right)\mapsto\frac{a}...
- 650
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Why do we need that the natural numbers are the intersection of all inductive sets in the reals?
I'm trying to understand this definition. Actually I have seen a lot of questions that describes this topic from the set theoretical point of view, but I would to ask it again from the point of view ...
- 2,437
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2
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How is ($\mathbb{Z}\setminus\mathbb{Q}$) a subset of $\mathbb{N}$?
I do not understand why the set ($\mathbb{Z}\setminus\mathbb{Q}$) is a subset of $\mathbb{N}$. $\mathbb{Q}$ extends the $\mathbb{Z}$ by adding fractions. So there cannot be an element in $\mathbb{Z}$ ...
- 21
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Is the infinite union of Cartesian products of countable sets, countable? [duplicate]
We know for $A_1, A_2, \cdots, A_n$, where each $A_i$ is countable, that $A_1 \times A_2 \times \cdots \times A_n$ is countable.
We also know that $\mathbb{N}$ is countable.
Let $N_i = \mathbb{N} \...
- 775
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3
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165
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Let set A is subset of set $\{{1,2,3,...,100}\}$ which has $55$ different elements.
Let set $A \subset\{{1,2,3,...,100}\}$ with $55$ distinct elements.
Prove that there exist two elements in $A$ which have a difference of 10.
I understand that the common thing is that we take more ...
- 425
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4
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365
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$\mathbb{N} ⊇ A_1 \supset A_2 \supset A_3 \supset \cdots$ but $\bigcap_{n=1}^∞ A_n$ is infinite?
What is an example of an infinite intersection of infinite sets is infinite?
I know that the intersection of infinite sets does not need to be infinite. However, I am seeking for an explicit example ...
- 51
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What are the finite and co-finite subsets of the set of Natural Numbers?
Let W be the set of Natural Numbers. What are all of the finite and co-finite subsets of W? It seems that all of the finite subsets of W will be inside of the power set of W, which we know will be ...
- 179
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Is there a bijective function $f: \Bbb Z \to \Bbb N$ that involves only elementary arithmetic and no piecewise functions?
As the title suggests, I'm looking for a function $f : \Bbb Z \to \Bbb N$ that satisfies the following:
$$
\forall y \in \Bbb N, \exists! x \in \Bbb Z : y = f(x)
\\
\therefore\quad
\Bbb N = \left\{ f(...
- 227
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142
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Show that the cardinality of ($X$ ∪ {$x$}) is equal to the cardianlity of ($X$)+$1$
Let $X$ be a finite set, and let $x$ be an object which is not an element of $X$. Then $X$ ∪ {$x$} is finite and #($X$ ∪ {$x$}) = #($X$)+$1$. Note that #-here means cardinality.
Suppose the cardinality ...
- 1,689
1
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648
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Show that the finite subsets of the natural numbers are bounded.
Let $n$ be a natural number, and let $f : ${$i ∈ N :1≤ i ≤ n$}→$N$ be a function. Show that there exists a natural number $M$ such that $f(i) ≤ M$ for all $1 ≤ i ≤ n$.Thus finite subsets of the natural ...
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