Exercise 107.15.2. Show the following

the complement of a constructible set is a constructible set,

a finite union of constructible sets is a constructible set,

a finite intersection of constructible sets is a constructible set, and

any constructible set $E$ can be written as a finite disjoint union $E = \coprod E_ i$ with each $E_ i$ of the form $Z \cap \{ f \not= 0\} $ where $Z$ is an algebraic set and $f$ is a polynomial.

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