Cardinality of set of all monotone increasing functions between $\Bbb N$

Question

Let $X =\{f:\Bbb N \rightarrow \Bbb N\mid f \text{ is monotone increasing function}\}$

Then want to find the $card(X)$

First what I know is function $f$ is equal to the monotone increasing sequence that is defined on $\Bbb N$

and I want to find any set which can be mapped from X to the set which would be injective and also another set which can be mapped from that set to X which would be injective.

Any suggestion for those kind of sets?

score 3 · Answer 1 · answered Jun 07 '17 at 05:11

3

Do you know how many functions there are $\Bbb N \to \Bbb N$? For any $f$ in your set, we can have $g(1)=f(1), g(n)=f(n)-f(n-1)$ This is a bijection between the $f$s and $g$s and I claim the $g$s are all functions $\Bbb N \to \Bbb N$. The bijection also says the set of $g$s could be your $X$.

answered Jun 07 '17 at 05:11

Ross Millikan

374,822

So you mean.. there's no way to specify the cardinality of my set X? – delog Jun 07 '17 at 05:37
Sure there is. The set of functions from $A$ to $B$ has cardinaltiy $|B|^{|A|}$ so your set $X$ has cardinality $|\Bbb N|^{|\Bbb N|}=\mathfrak c$ – Ross Millikan Jun 07 '17 at 13:47
is $\frak c = 2^{\aleph_0}$? – delog Jun 07 '17 at 13:48
Yes, it is. It is the cardinality of the reals. – Ross Millikan Jun 07 '17 at 13:53

score 2 · Answer 2 · answered Jun 07 '17 at 05:03

2

Think about the functions that have this property: $$f(n)\in\{2n-1,2n\}$$ for all $n$. Are these automatically increasing? How many of them are there?

answered Jun 07 '17 at 05:03

Angina Seng

158,341

I could easily imagine one such as f(n) = 2n-1 when n is odd number and f(n) =2n when n is even number. but how many of all f(n) that satisfies that property? .. that's tough – delog Jun 07 '17 at 05:06

Cardinality of set of all monotone increasing functions between $\Bbb N$

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