0
votes
3answers
52 views

Are the following sets countable?

I'm trying to determine if the following sets are countable: (a) $\mathbb{Z}^{[0,1]}, (b) [0,1]^{\mathbb{Z}}, (c) \mathbb{Z}^{\mathbb{Z}}$, (d) the set given by functions $f:\mathbb{Z}\to\mathbb{R}$ ...
0
votes
1answer
41 views

Find the cardinal of the set of all infinite sequences of $0,1,-1$ such that each sequence contains each digit at least once - Check my answer

As the title says, we are asked to find the cardinal of the set of all infinite sequences made from the digits $0,1,-1$ such that each sequence contains each digit at least once. My answer I solved ...
3
votes
2answers
125 views

How many functions $f^m(n) = n$ over $\mathbb{N}$?

I got a task that i have problem with. I have to find how many functions are there that satisfies $$f^m(n) = n$$ for some $m > 0$. So, i came up with an idea. How many functions there are for ...
1
vote
1answer
40 views

cardinality of $S_{\mathbb{N}}$

When I am proving something, I got a doubt. what is the cardinality of $S_{\mathbb{N}}$, the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$? I hope it is countable, because that will make my ...
1
vote
2answers
206 views

How many total order relations on a set $A$?

Let's define a set $T_A$ which is the set of all total order relations on $A$. This set is a subset of the set of all $2$-adic relations on $A$: $$T_A \subset \mathcal P(A^2) $$ 1-Which is the ...
0
votes
2answers
141 views

Countable or uncountable

(1) $C$ is the set of all circles $C(z,r)$ with $z\in\mathbb{Q}\times\mathbb{Q}$ and $r\in\mathbb{Q}^+$. What is the cardinality of $C$? (2) Let $S$ be the set of all sequences ...
1
vote
0answers
79 views

Closing a subcategory under finite colimits by transfinite induction

Let $\mathcal{C}$ be a locally small category with all finite colimits, and let $\mathcal{A}$ be a small full subcategory. I wish to prove the following: Proposition. There exists a full subcategory ...
2
votes
2answers
61 views

Cocountable fibers

Let $C$ be an uncountable set. Can we construct a set $A \subseteq C^2$ such that it has a cocountable number of cocountable horizontal fibers, and a cocountable number of countable vertical fibers?