@BAYMAX If $A\ge B\ge C$ (groups) then $[A:B]$ divides $[A:C]$. In fact, there is a $[B:C]$-to-$1$ map between coset spaces $A/C\to A/B$ given by $aC\mapsto aB$.
Anyway: in responce to the things I was thining about; I was pretty sure that I would need something along the lines of:
There exists (a thing) such that (a thing) is in X (where X is a set). Howver I need to write that in such a way that putting the thing in X puts a piece of the set in X and not a single element.
suppose we have a set. THe powerset of it would be a set for which all its elements are in the first set. However it is not a requirement that all the elements in the first are in the second.
Which would read: $P(X)\equiv \exists X \forall Z [Z\in X]$
Sorry I noticed that the first part was wrong after I posted but I wanted to know if the formate was right. If the P() part was a statement that defined the properties of the powerset; would it be correct?
appart from the P() part; is the syntax correct for defining something?
Say I had a statement that would only be true for a set which contained all the subsets of some other set. If I substituted that statement for P(); would the rest of the syntax be correct for defining the powerset?
the powerset of something. I suppose I should write $\mathscr P(x)$ I just didnt want to use x beucase I thought i might run out of symbols. In the future this is what i will use.
I have a question about induction: Generally, we prove some property for $n = 1$, then we take as hypothesis that it holds for some $n$ (for example, $a(n) = b(n)$ Eq1.). And finally we want to prove the same property holds for $n+1$ (for example, $a(n+1) = b(n+1)$ Eq2). My question is: It is rigorous use the Eq2? For example, we could say $a(n)-a(n+1) = b(n) - b(n+1)$.
I don't think this is rigorous, since, actually, we don't even know what relation there is between $a(n+1) = b(n+1)$. It shouldn't be there an $=$, it should be something like $a(n+1) \square b(n+1)$.
would that read $\mathscr P (X)=Y:$<insert statement containing X and Y> ? @DHMO
it seems that we would be anding our object with the statement in the above
I have made another attempt with the following. I have spotted a problem with my attempt but so far I am not sure how to fix it: $\mathscr P(X)=Y: (Y\leftrightarrow \forall a(a\subseteq X))$
The problem with it is that writing $\forall a(a\subseteq X)$ will not produce a set containing all the subsets.(I think it will instead produce the same set we started with (if the syntax here is even allowed)).
@DHMO I belive I can fix this problem now by addopting the syntax from the Axiom. If I instead write: $\mathscr P(X)=Y:\forall a(a\in Y\leftrightarrow (a\subseteq X))$ I will ensure that the elements of Y are subsets of X
I figured out the first part myself; but yes I had to cheat in the end.
I understand the formulation of the final answer; but there are some things in your last cpmment that I dont
First ill will explain why the answer works; then I will ask about your comments.
The difference between P(X)=Y≡(Y≡∀a(a⊆X)) and P(X)=Y≡∀a(a∈Y≡(a⊆X)) is that in the first one; the "∀a(a⊆X)" part will be true for a set that contains only the elements of X. this is not what we want beucase the intent is to create something that is only true for the power set of X
in the second one: we have created a set Y that has elements which are subsets of X. This is true for all a so the set will contain all the subsets
I have always taken it as the variable somehow binding itself; while a bounded one would be bound by some external factor. But I don't have a hard definition of bound variable and I have avoid questions that use the term on SE.
@DHMO Does my explanation of why it works make sense?
@DHMO That was one of the questions I was meaning to ask about your comments: What is the definition of $\equiv$? I have one for $\iff$ which is $A\iff B$ is $AB \lor (\lnot A)(\lnot B)$ but I dont have one for $\equiv$.
I think if I did I would be able to understand the difference between objects and statements.
Suppose $\sum_{n=0}^{\infty} \frac {B_n} {n!} \cdot z^n$ is a convergent power series whose pointwise limit is $f(z) = 1$ when $z = 0$ and $f(z) = \frac {z} {e^z - 1}$ when $z \ne 0$. How do you show that the radius of convergence is $2 \pi$?
If we cant do that; what are we doing when we write: $\mathscr P(X)=Y\equiv \forall a(a\in Y\equiv (a\subseteq X))$ $\mathscr P(x)$ is an object, and the rightmost side is a statement. Have we not connected them just now using it?
Ah now I see it. $\mathscr P(X)$ is an object; $Y$ is an object. $\mathscr P(X)=Y$ is a statement, and $[\mathscr P(X)=Y]\equiv A$ is a statement. In this we never connect objects to statements. Just object to object and statement to statement.
I have an answer without objects in it so I think it is incomplete for now. But The stataement part of the answer would look like: $\forall X\forall Y(\exists Z[(X\cup Y)\leftrightarrow Z])$
Somewhere before or after that I belive I need to make a statement that contains objects and set that statement to equal what I just wrote.
Really? In propositional logic we have statements that are represented by a single letter. And those statements can be continualy broken down until we arrive at atomic ones.
I have used it in excerscises but I have never answered a question on SE that involved it. This is becuase I have been afraid of using anything that I dont understand.
I have looked at it definition on wikipedia but it seems to define it through examples
and from the examples all I can gather is it takes a bunch of statements: Then AND's them togther. In this way the truth value of a statement containing $\forall$ will only be true when all the statements it AND's togther are true.
After what we have discused I don't belive I can write anything (althogh I did before). I know some of the rules for for all and can find logical equivilances for it. but thats it.
I just happen to like the formal stuff to be honest. But from what I have seen so far jumping to the formal stuff first has been a terrible way to learn it.
any good reference for the proof of this theorem "Let $H$ be a normal $p$ -subgroup of group $G$ ,then $H$ is contained into each Sylow $p$ subgroup of $G$",i could not find after googling ??
does someone now the difference between $|Map_{surj}[(1,k),(1,n)]=n! \cdot \stirling{n}{k}|$ and $|Part_k(X)|= \stirling{n}{k}$? I know, the calculation is different. My problem is that I do not know in which task I have to use which one. For example, if I have 10 Students and make 3 Tutoriums where no one is empty, which to choose?
I know that the process of making the power set involves re-writing each element of the first: giving us n elements; followed by taking one element away from the original set and and writing it again; giving us n +1 elements. We do that step n times. And we do those n times n times to get each combination.
I'm just not sure how to write that out in a proof
we add the empty set, re-write the original set; then the remove one element n times and add those new sets. Then we remove 2 elements n times and add those.
so it seems with add n things n-1 times. Then add two things
