Independent random variables implies independent $\sigma$-algebras

Question

Let $Y_1,...,Y_n,Y_{n+1}$ be independent random variables and $h_1,...,h_n$ measurable.

Question 1: How can I show that the $\sigma$-algebras $\sigma(Y_1,...,Y_n)$ and $\sigma(Y_{n+1})$ are independent?

Question 2: How can I show that $\sigma(h_1(Y_1,...,Y_n),...,h_n(Y_1,...,Y_n))$ and $Y_{n+1}$ are independent?

Is there anyway to prove the two problems?