not_a_math_guy

As far as I understand, $X$ is a member of $\mathcal{T}$ which is a topology on some other set..?

I am reading Hogg, McKean and Craig book for Introduction to mathematical statistics 6th ed. There are 2 two definitions there for continuous random variable (R.V): 1- R.V for which space is continuous (for e.g. space is real numbers) are continuous R.V. 2- R.V is continuous if its cumulative distribution function is continuous. Which one is correct or both are equivalent in some way?

$f(2n) = n$ is the bijection where $f:A \to \mathbb{N}$ and $A$ is set of all even natural numbers.

@Jakobian We can construct. Let me try for a explicit formula.

@Jakobian I think countable as there exists a injection from the set to $\mathbb{N}$. I can label each element of the set by a natural number.

@Jakobian I realize I do not have the needed understanding of the concept. I am trying to build one...

@Jakobian For countable sets, I can map the elements of the set to $\mathbb{N}$ and count them, it will represent the cardinality (is this correct?) for uncountable sets I am not sure what does cardinality means?

@VladimirLysikov because $A'$ can be equal to $A$. Right?

Wait, if $A$ is countably infinite set and $A'$ is an infinite subset of $A$ then also $|A'| \leq |A|$ is false.

@VladimirLysikov In my original post, $A' \subset A$ then $|A'| \leq |A|$ (for countable sets) and $B \subset P(A)$ but I think I get what you want to say.

@Jakobian I am thinking on it. will tell.

@Jakobian I'll give an example of what I meant to say: $A = \{a,b\}$ then $P(A) = \{\phi, \{a\}, \{b\}, \{a,b\}\}$ hence cardinality of $A$ can be seen as number of singleton subsets of $A$ in $P(A)$ and $P(A)$ will have more elements than singleton subsets of $A$ that is obvious...

@Jakobian yeah I realize it now while thinking for making my above argument more precise..

@Jakobian Is my argument not mathematically precise or its wrong?

@Jakobian If $A$ is some countable set then $P(A)$ will contain singleton subsets of $A$ additionally it will also contain subset with cardinality $2, \dots$ and $\phi$. So the cardinality of $P(A)$ is greater than $A$ or any subset of $A'$

@Jakobian Cardinality of $P(A)$ cannot be equal to some subset $A'$ of $A$.

What I think is, this proof will work for countable sets $A$ but for uncountable sets I am not sure if it's correct. Am i correct?

Proof: "Suppose for contradiction that there exists a surjection $f: A \to P(A)$. This means that for each $B \subset P(A)$, there exists $a \in A$ such that $f(a) = B$. Also, $P(A)$ is injective to some subset $A'$ of $A$ (we can construct this by taking minimum of all $a$'s for which $f(a) = B$). Now, $f$ is surjection as well as injection hence $|A'| = |P(A)|$ which is a contradiction. "

Proof verification of cantor's theorem: For any set $A$, there does not exist any surjection from $A$ to $P(A)$.

ohk. I think its better if this is mentioned as a footnote on wikipedia. Just so first comers will not get confused. Anyways, thanks.

There seems to have inconsistency in munkres and wikipedia

Munkres defines order relation as something which is non-reflexive (along with other properties) and provides the name "simple order" or "linear order" for it. While on $\href{en.wikipedia.org/wiki/Total_order}{wikipedia}$ it is mentioned that "simply/linearly" ordered set terms can be used for "totally ordered" set but their definition of totally ordered set contains reflexivity hence it can not be a "order relation".

@Jakobian oh I see.

@Jakobian is there a time limit for editing the post? I can't edit now. Btw thanks for clarifying the doubt.

Hello everyone, I just have a clarification that about what exactly are the elements of the $G/H$? Are they cosets $G/H = \{g_1H, g_2H, \dots \}$ where $g_i \in G$ or $ G/H = \{g__1h_1, g_1h_2, \dots, g_2h_1, g_2h_2, \dots \}$ where $h_i \in H$?