Detemine cardinality of $ \{f \mid f: \mathbb{N} \rightarrow [0,1] \}$.  We can show ran$(f) \neq [0,1] $  by a diagonal argument. But I am not sure how to use this to determine the cardinality.
9911
I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, since the cardinality of all rational sequences is the same as the cardinality of reals, and it seemed fairly intuitive to me that if cardinality of a set $A$ is strictly greater than the cardinality of the set $B$, then cardinality of $A^{\mathbb{N}}$ should be strictly greater than cardinality of $B^{\mathbb{N}}$. It turns out to be false.  Some technical answers have appeared in this forum elsewhere but I do not understand them. As I am not an expert in this topic, could some one explain me in simple terms why this is happening?  Also is the cardinality of all functions from reals to reals also the same as the cardinality of reals?
15141
I need to find $|GL_2(F_p)|$, I am very very uncomfortable in counting, so please help here I proceed,  $\begin{pmatrix}a&amp;b\\c&amp;d\end{pmatrix}$ be such an invertible matrix, so we need and sufficient that $ad\neq bc$, so if i chose all non zero $a,b,c$ arbitrarily and chose $d\neq a^{-1}bc$ and form matrices then the cardinality   will be $(q-1)^3\times (q-2)$, well now if one of $a,b,c,d$ be zero then other three can be anything nonzero so in this case cardinality will be $4(q-1)^3$, so finally if two entries are $0$ then they must be opposite to each other i.e diagonally, but other two must be nonzero, so cardinality in that case $2(q-1)^2$, so total number is just sum of three cases.
44114
Question:  Prove that the interval $[0,2)$ and $[5,6) \cup [7,8)$ have the same cardinality by constructing a bijection between the two sets. You can add a graph to support your argument.   I tried graphing a function with domain $[0,2)$  and target space $[5,6) \cup [7,8)$ but I can't come up with an explicit formula. Will the function be linear?
65235
I am stuck on this problem:  Define $f: N\times N \rightarrow N$ by   $f(i,j)=\frac{(i+j-1)(i+j-2)}{2}+i$.   How do you prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent?  Work: First I tried to prove that $f$ is injective by setting $f(i,j)=f(x,y)$. However, from here I am stuck and I cannot reduce it properly. My work really does not get anywhere significant and its really messy.
75969
I came across this long proof on this site:    But I would like to know whether my direction can work.  Say we want to find the cardinality of all equivalence relations in $\mathbb{N}$. Since it is a subset of all relations in $\mathbb{N}$, I conclude it has a cardinality smaller or equal to $\aleph$. Now, define an injective function from $P(\mathbb{N})$ to the set of equivalence relations by matching each subset of $\mathbb{N}$ with the identity relation (which is an equivalence relation in $\mathbb{N}$.  Therefore the cardinality of all equivalence relations in $\mathbb{N}$ is greater or equal to $\aleph$ and using CSB we get the desired result.  Seems legit?
45887
Let $A$ be a set with card($A$)=$a$. What is the cardinal number of the set of countably infinite subsets of $A$?   I see that this problem is equivalent to finding the cardinal number of the set of injective functions from $\mathbb{N}\rightarrow{A}$. I also know that the cardinal number of the set of bijections from $A\rightarrow{A}$ is $a^{a}$.   Hints and general heurisitcs would be greatly appreciated.
48867
Proving that $[0,1)$ and $(0,1)$ have the same cardinality (without assuming any previous knowledge) can be done easily using Cantor-Bernstein theorem.  However I'm wondering if someone can build an explicit bijection between these sets.  It's easy to build a bijection between $(0,1)$ and $\mathbb R$, so a bijection from $[0,1)$ to $\mathbb R$ will also fit the bill.
83992
I learned that the set of all one-to-one mappings of $\mathbb{N}$ onto $\mathbb{N}$ has cardinality $|\mathbb{R}|$.  What about surjective functions and bijective functions?
61719
The book Theory of Functions of a Real Variable by I. P. Natanson, proves that a denumerable or finite union of pairwise disjoint sets of cardinality $c$ has cardinality $c$.  The proofs given in the book are fairly easy, using the axiom of choice. But then it is left as an exercise a generalization of the above theorems for not necessarily pairwise disjoint sets.  I have searched all over the web, but all the proofs found so far are for pairwise disjoint sets.  The proof given in the book relies on the fact that each set $A_k$ can be matched with a set like $[a_{k-1},a_k)$, with $a_k$ point of a partition of $[0,1]$, such that $\bigcup_{k=1}^{\infty\lor n}A_k=[0,1)$. So it is easy to guess the bijective function, but I can't generalize this proof.  Any help is highly appreciated. Thanks and regards.
46950
Initially assume ZFC.  Let $\binom{\kappa}{\lambda}$ denotes $\left|[\kappa]^{\lambda}\right|$ where $[\kappa]^{\lambda}$ is the collection of all subsets of $\kappa$ with cardinality $\lambda$. That is, the number of subsets of $\kappa$ of size $\lambda$.  Let $\kappa$ be any infinite cardinal number. Then it is easy to see that   $\binom{\kappa}0=1$ $\binom{\kappa}n=\kappa$ for all positive natural number $n$.  $\binom{\aleph_0}{\aleph_0}=\beth_1$ $\binom{\kappa}{\lambda}\le \kappa^\lambda$ when $1\le \lambda \le \kappa$   3 is due to $\beth_1=\sum_{\lambda=0}^{\aleph_0}\binom{\aleph_0}{\lambda}$ and others besides $\binom{\aleph_0}{\aleph_0}$ sums $\aleph_0$(since countable infinite countable infinites sums to countable infinite)  $\left({}^{\kappa}_{\lambda}\right)\le \kappa^\lambda$ is due to an injection can be made from $[\kappa]^{\lambda}$ into $\kappa^{\lambda}$(maps $\{a_i|i&lt;\lambda\}$ to $\langle a_i\rangle_{i&lt;\lambda}$ where $a_j&lt;a_k$ iff $j&lt;k$)  I hypothesize that   $$\binom{\kappa}{\lambda}=\begin{cases} \kappa^\lambda,&amp;\lambda \le \kappa \\ 0,&amp; \lambda&gt;\kappa  \end{cases}$$  It holds when $\kappa=\aleph_0$, but hard to induct for larger cardinals without assuming GCH. So I need some helps.
94508
Some very wise people here have just told me that  $\aleph_0 = \mathbb{N}$, i.e. that the cardinality of the set of natural numbers is just the set of natural numbers itself. Is this now the general consensus in mathematics, real analysis in particular? Or have I got completely the wrong end of the stick as usual?
92322
Given a set of N elements, compute the number of subsets whose cardinalities are congruent to 0, 1 and 2 modulo 3 respectively.  Any hints would be appreciated.  Thanks!
42314
I am trying to prove that, given a collection $\{S_i:i \in \mathbb{N}\}$ of infinite sets which all have the same cardinality, that $|S_1 \cup S_2 \cup \cdots|=|S_1|$. WLOG, I'm going to assume the $S_i$ are disjoint.  My thought was to let each $f_i:S_1 \to S_i$ be a bijection and to define $g: S_1 \times \mathbb{N} \to S_1 \cup S_2 \cup \cdots$ by $g(x,n)=f_n(x)$. I see that $g$ is a bijection.   So, the problem is reduced to "merely" proving that $|S_1 \times \mathbb{N}|=|S_1|$. I say "merely" with tongue in cheek because it is by no means obvious to me how to do it.  If $S_1$ is countable, I know how to proceed by the standard diagonal argument used to show $\mathbb{Q}$ is countable. But if $S_1$ is uncountable, I don't know what to do, even though it seems intuitive to me that crossing $S_1$ with $\mathbb{N}$ won't make the set "bigger" since it doesn't make it bigger in the countable case.
25677
I have to prove/disprove this:     If $\det(A+X) = \det(B + X)~ \forall X \in M_{n \times n} (\mathbb F) \rightarrow A = B$   I believe it is true but I can not think of a direct way to prove it. Any help is appreciated!
83697
I have two questions, actually. The first is as the title says: how do we know there exists an infinite cardinal such that there exists no other cardinals between it and $ \aleph_0 $? (We would have to assume or derive the existence of such an object before we label it something like $ \aleph_1 $.)  My second question is, can we say for certain if there's any limit to the number of cardinals existing between $ \aleph_0 $ and continuum (i.e. $ 2^{\aleph_0} $)? I mean, how could we know that there&#39;s not an infinite number of cardinals between the two - perhaps even more than $ \aleph_0 $?
67114
I want to prove that any open interval $(a,b)$ has the same cardinality of the real numbers: $|(a,b)| = |\Bbb R|$.   Do I have to find an function to prove it? Or is there a theorem to prove it easier? or any idea?
56222
Show that a ring $ R $ with exactly $ n $ zero divisors (different from $0$) has cardinality atmost $ (n+1)^2 $.    I have shown that annihilator of any element among the zero divisor is a subset of the zero divisor which proves it is finite. now i think i have to show that it's coset space is finite.please help me.