properties of ideals in $K[x_1,\ldots ,x_n]$

I'm trying to convince myself of the 4 following facts:

  1. If $X \subseteq Y \subseteq A^n_{k}$ then $I(Y) \subseteq I(X)$
  2. If $J \subseteq K[x_1,\ldots,x_n]$ is an ideal then $J \subseteq I(V(J))$
  3. If $X \subseteq A^n_{k}$ then $X \subseteq V(I(X))$
  4. $X=V(I(X))$ if and only if $X$ is an algebraic set

My attemps to prove them:
1) Let $f \in I(Y)$ then $f(p)=0$ for all $p\in Y$. So since $X \subseteq Y$ we have $f(p)=0$ for all $p \in X$, so $f \in I(X)$
4) -> If $X=V(I(X))$ then X= V(something) which is an algebraic set.
<- suppose an algerbaic set, i.e $X=V(J)$. Then by (2) $J \subseteq I(V(J))$.
so $I(V(J))=I(X)$.
so $X \subseteq V(I(X)) \subseteq V(J)$ by (3). So $X=V(I(X))$.

Is this right? I use part 2 and 3 for part 4, but cannot convince myself algebraicly why they hold.

2022-07-25 17:47:17
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