All Questions
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questions
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proving the set of natural numbers is infinite (Tao Ex 2.6.3)
Tao's Analysis I 4th ed has the following exercise 3.6.3:
Let $n$ be a natural number, and let $f:\{i \in \mathbb{N}:i \leq i \leq n\} \to \mathbb{N}$ be a function. Show that there exists a natural ...
- 3,325
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1
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125
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What is the cardinality of non-singleton subsets of $\mathbb{N}$?
I am studying a course on ZF Set Theory (without the Axiom of Choice) and am currently looking at the cardinalities of infinite sets. One question that I came across is the following:
Determine the ...
- 4,331
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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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1
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Countability of a nonincreasing set
I have a function $f \in \mathbb{N}$ that is nonincreasing for $x,y \in \mathbb{N}$.
Now I have to prove that the set $$A:=\{f \in \mathbb{N}^{\mathbb{N}} | f\ \text{ is nonincreasing} \}$$ is ...
- 13
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How to prove that $N\setminus A$ is finite? [closed]
$A \subseteq \mathcal{R}(N)$ and given that (by inductive definition):
$N ∈ S$.
If $a \in R$, then $R \setminus \{a\} \in A$.
I need to prove that for each $A \in S$, $N\setminus A$ is finite. How ...
user567507
17
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2
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Is aleph-$0$ a natural number?
Would I be right in saying that $\aleph_0 \in \mathbb N$?
Or would it be a wrong thing to do?
user108343