Chapter 2, Problem 2.9STPE

For a finite set A, let N(A) denote the number of elements in A.

a. Show that $N\left(A\cup B\right)=N\left(A\right)+N\left(B\right)-N\left(AB\right)$

b. More generally, show that $N\left({\displaystyle \underset{i=1}{\overset{n}{\cap }}{A}_i}\right)={\displaystyle \underset{i}{\sum }N\left(A_i\right)}-{\displaystyle \underset{i<j}{\sum }N\left(A_i{A}_j\right)}+\mathrm{...}+{\left(-1\right)}^{n+1}N\left(A_1\mathrm{...}{A}_n\right)$