Cure

Say, in relation to my question a while ago:
Given $F(w,x,y,z)=(F_1,F_2, F_3)$ and I would like to express each $x,y,z$ in terms of $w$.

This would be possible if the determinant for the jacobian matrix with partial derivatives for $x,y,z$ were non-zero.

I have shown that it is zero, does that mean I can't express $x,y,z$ in terms of $w$?

Hello! I have a question regarding invertible functions. Say I have a function $f:\subset\mathbb{R}to\mathbb{R]$. If I find an inverse function $f^{-1}$ using some computations, and then I find the jacobian matrix of both $J_{f}$ and $J_{f^{-1}}$.

Should I expect $[J_f}]^{-1}=[J_{f^{-1}}]$ ?

I have a question about permutations. I'm trying to prove there is no permutation $\alpha$ such that $\alpha(123) \alpha^{-1}=(13)(578)$.

I think I would have to use that $\alpha(123) \alpha^{-1} = (\alpha(1)\alpha(2)\alpha(3))$.

Is the fact that $(13)(578)$ is a disjoint 2-cycle permutation and $(\alpha(1)\alpha(2)\alpha(3))$ is one-cycle enough to argue there is no $\alpha$ that satisfies the condition?

But once in formalized theory, can anything in regards of \in be said before going into the ZF axioms?

For example, I came across this theorem: \forall x \forall y [x=y \implies \forall z( z\in x \iff z\in y )]

I have what I think is trivial question, but I cannot get my head around it. I'm reading about on Foundations of Math, and it starts with set theory, giving the ZFC axioms.

One of the goals of the ZF restrictions on set is to avoid the formation of bigger sets that lead to contradictions, like the one given by Russell's paradox, which ultimately leads to A∈A therefore A∉A , and if A∈A follows A∈A. But why is it a contradiction?